Derniers messages sur Zeste de Savoirhttps://zestedesavoir.com/forums/2018-11-14T14:25:26+01:00Les derniers messages parus sur le forum de Zeste de Savoir.Connaître l'angle par rapport au premier point du vecteur, message #1930512018-11-14T14:25:26+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p193051<p>Ça fonctionne correctement maintenant avec mes derniers changements, mon soucis était que le résultat n’était pas optimisé avec cette fameuse condition pour la conversion.</p>
<p>orientationNSWE m’est donné, je ne peux pas modifier la valeur en entrée, à moins de faire la conversion sur elle mais ça déplacerait le problème ailleurs.</p>Connaître l'angle par rapport au premier point du vecteur, message #1930502018-11-14T13:58:36+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p193050<p>Ah, d’accord, j’avais mal lu/compris. C’est donc simplement l’angle exprimé en degré et dans le sens indirect.</p>
<p>Tu ne pourrais pas plutôt l’avoir dans le sens direct (donc 90° au Nord et 270 au Sud) ? Ce serait plus simple je pense. La conversion entre les deux est simple, il suffit de multiplier par -1 (modulo 360 si tu veux un résultat positif).</p>
<p>Mais au final je n’ai pas compris si tu étais arrivé à quelque chose de fonctionnel ou si tu avais encore des soucis.</p>Connaître l'angle par rapport au premier point du vecteur, message #1930452018-11-14T13:46:33+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p193045<p>orientationNSWE est mon angle sur l’axe Nord, Sud, Ouest et Est.</p>
<p>Si je regarde à l’est alors angle = 0°, si je regarde au sud 90°, ouest 180° et nord 270°.</p>
<p><strong>EDIT :</strong> Mon doigt a glissé, j’ai cliqué sur envoyer au lieu d’aperçu</p>Connaître l'angle par rapport au premier point du vecteur, message #1930432018-11-14T13:34:19+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p193043<p>Je ne comprends pas ce qu’est cette <code>orientationNSWE</code>, c’est une valeur à toi ?
Tu la décris comme un ensemble de quatre nombres, mais la manière dont tu l’utilises laisse penser que c’est un seul nombre, donc je suis perdu.</p>Connaître l'angle par rapport au premier point du vecteur, message #1930082018-11-13T14:33:45+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p193008<blockquote>
<p>Enfin je vois bien que tu veux convertir la valeur en degrés (est-ce réellement utile ?), mais pour le reste ?</p>
</blockquote>
<p>L’entrée <kbd>orientationNSWE</kbd> est en degrés je suis donc obligé de faire la conversion.</p>
<hr>
<p><code>getAngle(positionXY, destinationXY)</code> me donne l’angle du vecteur (donc de mon mouvement) sous le même format que orientationNSWE : E=0°, S=90°, W=180°, N=270°.</p>
<p><kbd>orientationNSWE</kbd> est l’angle absolu de mon regard (E=0°, S=90°, W=180°, N=270°)</p>
<p>En soustrayant l’angle du vecteur avec <kbd>orientationNSWE</kbd>, j’obtiens le delta qui représente la direction où regarder pour prendre ma photo. <img src="/static/smileys/heureux.png" alt=":D" class="smiley"> Où <code>angle=0</code> est le nez de l’appareil et <code>angle=180</code> ou <code>angle=-180</code> la queue, <code>angle=-90</code> ou <code>angle=90</code> le côté droit ou gauche.</p>
<hr>
<p>Pour ma condition :</p>
<div class="hljs-code-div"><div class="hljs-line-numbers"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></div><pre><code class="hljs language-text"> N=270
|
dx=1, dy=-1 | dx=1, dy=1
|
W=180-------------+-------------E=0 ou 360
|
dx=-1, dy=-1 | dx=1, dy=-1
|
S=90
</code></pre></div>
<p>J’ai dû inversé dy car le nord-sud est inversé dans l’entrée <kbd>orientationNSWE</kbd>.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929992018-11-13T13:25:21+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192999<p>Je ne comprends pas ce que tu essaies de faire à partir du résultat d'<code>atan2</code>. Enfin je vois bien que tu veux convertir la valeur en degrés (est-ce réellement utile ?), mais pour le reste ?</p>Connaître l'angle par rapport au premier point du vecteur, message #1929842018-11-13T09:58:24+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192984<p>Avant de faire ce sujet j’utilisais <code>Math.atan2</code> sans visualiser correctement que c’était un angle <em>universelle</em> avec mes tests je savais que c’était la fonction la plus adapté mais j’avais du mal à traiter le résultat que j’obtenais donc je me retrouvais avec un résultat final faux :</p>
<p>Mais en me penchant sur le sujet j’ai pu corriger grâce à cette visualisation. Merci. <img src="/static/smileys/smile.png" alt=":)" class="smiley"></p>
<p>J’ai pu réaliser la ligne 8 et 16 sachant que je comprenais vraiment le résultat d’atan2, on constate que ma solution n’est pas très mathématique :</p>
<div class="hljs-code-div"><div class="hljs-line-numbers"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></div><pre><code class="hljs language-js"><span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">getAngle</span>(<span class="hljs-params">a, b</span>) </span>{
<span class="hljs-keyword">const</span> x = b.x - a.x,
y = b.y - a.y,
dx = (x < <span class="hljs-number">0</span>) ? <span class="hljs-number">-1</span> : <span class="hljs-number">1</span>,
dy = (y < <span class="hljs-number">0</span>) ? <span class="hljs-number">-1</span> : <span class="hljs-number">1</span>,
atan2 = <span class="hljs-built_in">Math</span>.atan2(<span class="hljs-built_in">Math</span>.abs(y), <span class="hljs-built_in">Math</span>.abs(x));
<span class="hljs-keyword">let</span> axe; <span class="hljs-comment">// Chaque valeur de axe à +90 car sens horaire</span>
<span class="hljs-keyword">if</span> (dx === <span class="hljs-number">1</span> && dy === <span class="hljs-number">1</span>)
axe = <span class="hljs-number">90</span>;
<span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (dx === <span class="hljs-number">-1</span> && dy === <span class="hljs-number">1</span>)
axe = <span class="hljs-number">180</span>;
<span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (dx === <span class="hljs-number">-1</span> && dy === <span class="hljs-number">-1</span>)
axe = <span class="hljs-number">270</span>;
<span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (dx === <span class="hljs-number">1</span> && dy === <span class="hljs-number">-1</span>)
axe = <span class="hljs-number">360</span>;
<span class="hljs-comment">// /\ dy est inversé dans les 4 conditions</span>
<span class="hljs-comment">// pour que SUD soit égale à 90.</span>
<span class="hljs-keyword">return</span> axe - <span class="hljs-built_in">Math</span>.round(atan2 * <span class="hljs-number">180</span> / <span class="hljs-built_in">Math</span>.PI);
<span class="hljs-comment">// /\ soustraction pour le sens horaire</span>
}
<span class="hljs-function"><span class="hljs-keyword">function</span> <span class="hljs-title">correction180</span>(<span class="hljs-params">angle</span>) </span>{
<span class="hljs-keyword">if</span> (angle > <span class="hljs-number">180</span>) {
<span class="hljs-keyword">return</span> <span class="hljs-number">180</span> - angle;
} <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (angle < <span class="hljs-number">-180</span>) {
<span class="hljs-keyword">return</span> <span class="hljs-number">360</span> + angle;
} <span class="hljs-keyword">else</span> {
<span class="hljs-keyword">return</span> angle;
}
}
correction180(orientationNSWE - getAngle(positionXY, destinationXY))
</code></pre></div>
<p>On peut gérer la ligne 8 et 16 comme ci-dessous, mais ça complexifie le code.</p>
<div class="hljs-code-div"><div class="hljs-line-numbers"><span></span></div><pre><code class="hljs language-js"><span class="hljs-keyword">const</span> axe = <span class="hljs-number">90</span> * (<span class="hljs-number">1</span> + (dx * dy == <span class="hljs-number">-1</span>) + (dy == <span class="hljs-number">-1</span>) * <span class="hljs-number">2</span>);
</code></pre></div>
<p>Pour essayer dans la console web :</p>
<div class="hljs-code-div"><div class="hljs-line-numbers"><span></span></div><pre><code class="hljs language-js">(<span class="hljs-function">(<span class="hljs-params">dx, dy</span>) =></span> <span class="hljs-number">90</span> * (<span class="hljs-number">1</span> + (dx * dy == <span class="hljs-number">-1</span>) + (dy == <span class="hljs-number">-1</span>) * <span class="hljs-number">2</span>))(<span class="hljs-number">1</span>, <span class="hljs-number">-1</span>)
</code></pre></div>
<p>EDIT : Ou :</p>
<div class="hljs-code-div"><div class="hljs-line-numbers"><span></span></div><pre><code class="hljs language-js"><span class="hljs-keyword">const</span> axe = <span class="hljs-number">90</span> * (<span class="hljs-number">1</span> + (dx != dy) + (dy == <span class="hljs-number">-1</span>) * <span class="hljs-number">2</span>);
</code></pre></div>Connaître l'angle par rapport au premier point du vecteur, message #1929802018-11-13T08:54:49+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192980<p>Oui, ça te donne l’angle formé entre les vecteurs <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(1, 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span> et <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>3</mn><mo separator="true">,</mo><mn>3</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(3, 3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span></span></span></span></span>.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929562018-11-12T17:13:21+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192956<p>Si je comprends : <a href="https://developer.mozilla.org/fr/docs/Web/JavaScript/Reference/Objets_globaux/Math/atan2">https://developer.mozilla.org/fr/docs/Web/JavaScript/Reference/Objets_globaux/Math/atan2</a></p>
<p>Quand je fais</p>
<blockquote>
<p>Math.atan2(3,3)*180/Math.PI</p>
</blockquote>
<p>C’est comme si j’avais un triangle ABC : A(0;0) B(3;0) C(3;3) et il me donne l’angle de BAC ?</p>Connaître l'angle par rapport au premier point du vecteur, message #1929552018-11-12T16:37:48+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192955<p>Non la fonction <code>atan2</code> de la lib math, qui prend une ordonnée et une abscisse et renvoie un angle en radians. Elle est préférable à arccos ou arcsin car comprend tout le cercle trigonométrique, là où les deux autres se limiteront à des demi-cercles.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929522018-11-12T16:03:56+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192952<p>TAN ? <del>Mais je n’ai pas le côté opposé</del></p>
<p><strong>EDIT :</strong> <del>Si avec :</del> <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>tan2</mtext><mo>(</mo><mo>(</mo><msub><mi>B</mi><mi>x</mi></msub><mo>−</mo><msub><mi>A</mi><mi>x</mi></msub><mo>)</mo><mi mathvariant="normal">/</mi><mo>(</mo><msub><mi>B</mi><mi>y</mi></msub><mo>−</mo><msub><mi>A</mi><mi>y</mi></msub><mo>)</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">\text{tan2}((B_x - A_x)/(B_y - A_y))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="base"><span class="mord text"><span class="mord">tan2</span></span><span class="mopen">(</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mclose">)</span><span class="mord">/</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.05017em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mclose">)</span><span class="mclose">)</span></span></span></span></span></p>
<p><strong>EDIT2 :</strong> Mon précédent edit est faux. Je ne l’ai pas car il me manque un bout du côté adjacent, ma première explication était fause.</p>
<hr>
<blockquote>
<ul>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi mathvariant="normal">A</mi><mo>(</mo><mn>5</mn><mo separator="true">;</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{A(5;1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathrm">A</span><span class="mopen">(</span><span class="mord mathrm">5</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span></span></span></li>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi mathvariant="normal">B</mi><mo>(</mo><mn>1</mn><mo separator="true">;</mo><mn>4</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{B(1;4)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathrm">B</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">4</span><span class="mclose">)</span></span></span></span></span></span></li>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><msub><mi mathvariant="normal">O</mi><mtext>vni</mtext></msub><mo>(</mo><mn>2</mn><mo separator="true">;</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{O_{\text{vni}}(2;1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mathrm mtight">vni</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span></span></span></li>
</ul>
<p>Donc :</p>
<ul>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mover accent="true"><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow><mo>⃗</mo></mover><mo>(</mo><mo>−</mo><mn>4</mn><mo separator="true">;</mo><mn>3</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{\vec{AB}(-4;3)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9663299999999999em;"></span><span class="strut bottom" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">A</span><span class="mord mathrm">B</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">4</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">3</span><span class="mclose">)</span></span></span></span></span></span></li>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mover accent="true"><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">O</mi></mrow><mo>⃗</mo></mover><mo>(</mo><mo>−</mo><mn>3</mn><mo separator="true">;</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{\vec{AO}(-3;1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9663299999999999em;"></span><span class="strut bottom" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">A</span><span class="mord mathrm">O</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">3</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span></span></span></li>
</ul>
</blockquote>
<p>Je pense avoir trouvé :</p>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover><mspace width="0.277778em"></mspace><mo>∗</mo><mspace width="0.277778em"></mspace><mover accent="true"><mrow><mi>A</mi><msub><mi>O</mi><mtext>vni</mtext></msub></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow><mo>∗</mo><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">O</mi></mrow><mo>∗</mo><mtext>cos</mtext><mo>(</mo><mstyle><mover><mo><mtext>BAO</mtext></mo><mo>⌢</mo></mover></mstyle><mo>)</mo></mrow><annotation encoding="application/x-tex">\overrightarrow{AB}\;*\;\overrightarrow{AO_{\text{vni}}}=\mathrm{AB}*\mathrm{AO}*\text{cos}(\stackrel \frown \text{BAO})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.233678em;"></span><span class="strut bottom" style="height:1.483678em;vertical-align:-0.25em;"></span><span class="base"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mspace thickspace"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mspace thickspace"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">vni</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathrm">A</span><span class="mord mathrm">B</span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathrm">A</span><span class="mord mathrm">O</span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord text"><span class="mord">cos</span></span><span class="mopen">(</span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.233678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord text"><span class="mord">BAO</span></span></span></span></span><span style="top:-3.88333em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⌢</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></div>
<blockquote>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover><mspace width="0.277778em"></mspace><mi mathvariant="normal">.</mi><mspace width="0.277778em"></mspace><mover accent="true"><mrow><mi>A</mi><msub><mi>O</mi><mtext>vni</mtext></msub></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mrow><mo fence="true">(</mo><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>4</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>3</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mi mathvariant="normal">.</mi><mrow><mo fence="true">(</mo><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>3</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>)</mo><mo>=</mo><mo>−</mo><mn>4</mn><mo>∗</mo><mo>(</mo><mo>−</mo><mn>3</mn><mo>)</mo><mo>+</mo><mn>3</mn><mo>∗</mo><mn>1</mn><mo>=</mo><mn>1</mn><mn>5</mn></mrow><annotation encoding="application/x-tex">\overrightarrow{AB}\;.\;\overrightarrow{AO_{\text{vni}}} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}.\begin{pmatrix} -3 \\ 1 \end{pmatrix}) = -4 * (-3) + 3 * 1 = 15</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.45em;"></span><span class="strut bottom" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="base"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span><span class="mspace thickspace"></span><span class="mord">.</span><span class="mspace thickspace"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">vni</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">4</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">3</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">3</span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mord">5</span></span></span></span></span></div>
</blockquote>
<p>Avec :</p>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>B</mi><mo>=</mo><msqrt><mrow><mn>4</mn><mo>∗</mo><mn>4</mn><mo>+</mo><mn>3</mn><mo>∗</mo><mn>3</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mn>2</mn><mn>5</mn></mrow></msqrt><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">AB = \sqrt{4*4+3*3} = \sqrt{25} = 5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.956095em;"></span><span class="strut bottom" style="height:1.081665em;vertical-align:-0.12556999999999996em;"></span><span class="base"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9144300000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">4</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">4</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span></span></span><span style="top:-2.8744300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,
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s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.12556999999999996em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span><span class="mord">5</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,
-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,
35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,
-221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467
s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422
s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">5</span></span></span></span></span></div>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>O</mi><mo>=</mo><msqrt><mrow><mn>3</mn><mo>∗</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>∗</mo><mn>1</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mn>1</mn><mn>0</mn></mrow></msqrt><mo>=</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>1</mn><mn>6</mn><mn>2</mn></mrow><annotation encoding="application/x-tex">AO = \sqrt{3*3+1*1} = \sqrt{10} = 3.162</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.956095em;"></span><span class="strut bottom" style="height:1.081665em;vertical-align:-0.12556999999999996em;"></span><span class="base"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9144300000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span></span></span><span style="top:-2.8744300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,
-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,
35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,
-221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467
s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422
s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.12556999999999996em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">1</span><span class="mord">0</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,
-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,
35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,
-221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467
s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422
s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">3</span><span class="mord">.</span><span class="mord">1</span><span class="mord">6</span><span class="mord">2</span></span></span></span></span></div>
<p>Donc :</p>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mn>5</mn><mo>=</mo><mn>5</mn><mo>∗</mo><mn>3</mn><mi mathvariant="normal">.</mi><mn>2</mn><mo>∗</mo><mtext>cos</mtext><mo>(</mo><mstyle><mover><mo><mtext>BAO</mtext></mo><mo>⌢</mo></mover></mstyle><mo>)</mo></mrow><annotation encoding="application/x-tex">15=5*3.2*\text{cos}(\stackrel \frown \text{BAO})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.233678em;"></span><span class="strut bottom" style="height:1.483678em;vertical-align:-0.25em;"></span><span class="base"><span class="mord">1</span><span class="mord">5</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">5</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mord">.</span><span class="mord">2</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord text"><span class="mord">cos</span></span><span class="mopen">(</span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.233678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord text"><span class="mord">BAO</span></span></span></span></span><span style="top:-3.88333em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⌢</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></div>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mstyle><mover><mo><mtext>BAO</mtext></mo><mo>⌢</mo></mover></mstyle><mo>)</mo><mo>=</mo><mn>1</mn><mn>5</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mn>2</mn><mn>5</mn><mn>0</mn></mrow></msqrt></mrow><annotation encoding="application/x-tex">\mathrm{cos}^{-1}(\stackrel \frown \text{BAO}) = 15/\sqrt{250}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.233678em;"></span><span class="strut bottom" style="height:1.483678em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord"><span class="mord mathrm">c</span><span class="mord mathrm">o</span><span class="mord mathrm">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.233678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord text"><span class="mord">BAO</span></span></span></span></span><span style="top:-3.88333em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⌢</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mord">5</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span><span class="mord">5</span><span class="mord">0</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,
-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,
35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,
-221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467
s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422
s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"></span></span></span></span></span></span></span></span></div>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi></mrow><mo>(</mo><mn>1</mn><mn>5</mn><mi mathvariant="normal">/</mi><msqrt><mrow><mn>2</mn><mn>5</mn><mn>0</mn></mrow></msqrt><mo>)</mo><mo>=</mo><mn>1</mn><mn>8</mn><mi mathvariant="normal">.</mi><mn>4</mn><mn>3</mn></mrow><annotation encoding="application/x-tex">\mathrm{arccos}(15/\sqrt{250}) = 18.43</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.956095em;"></span><span class="strut bottom" style="height:1.206095em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathrm">a</span><span class="mord mathrm">r</span><span class="mord mathrm">c</span><span class="mord mathrm">c</span><span class="mord mathrm">o</span><span class="mord mathrm">s</span></span><span class="mopen">(</span><span class="mord">1</span><span class="mord">5</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.956095em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span><span class="mord">5</span><span class="mord">0</span></span></span><span style="top:-2.916095em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,
-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,
-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,
35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,
-221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467
s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422
s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.08390500000000001em;"></span></span></span></span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mord">8</span><span class="mord">.</span><span class="mord">4</span><span class="mord">3</span></span></span></span></span></div>Connaître l'angle par rapport au premier point du vecteur, message #1929512018-11-12T15:52:56+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192951<p>Ton calcul d'<span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>O</mi></mrow><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec{AO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9663299999999999em;"></span><span class="strut bottom" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="base"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.02778em;">O</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span></span></span></span></span> n’est pas bon, A et O ont la même ordonnée normalement.</p>
<p>Pour la suite tu peux en effet déterminer le cosinus de l’angle OAB à partir du produit scalaire. Et tu peux avoir le sinus avec le déterminant.
Oriente-toi alors vers la fonction <code>atan2</code> qui est la meilleure manière d’obtenir l’angle à partir de ces deux valeurs.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929502018-11-12T15:47:02+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192950<p>Je n’ai pas trouvé de documentation liée à Gamma/γ.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929492018-11-12T15:30:50+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192949<p>Tu connais la longueur de tes vecteurs. Ce que tu cherches, c’est <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>cos</mi><mo></mo><mo>(</mo><mi>γ</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\cos(\gamma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.05556em;">γ</span><span class="mclose">)</span></span></span></span></span> <img src="/static/smileys/clin.png" alt=";)" class="smiley"></p>Connaître l'angle par rapport au premier point du vecteur, message #1929482018-11-12T15:22:52+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192948<p>J’ai improvisé un autre essai, c’est faux <img src="/static/smileys/triste.png" alt=":(" class="smiley"></p>
<ul>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi mathvariant="normal">A</mi><mo>(</mo><mn>5</mn><mo separator="true">;</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{A(5;1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathrm">A</span><span class="mopen">(</span><span class="mord mathrm">5</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span></span></span></li>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi mathvariant="normal">B</mi><mo>(</mo><mn>1</mn><mo separator="true">;</mo><mn>4</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{B(1;4)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord mathrm">B</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">4</span><span class="mclose">)</span></span></span></span></span></span></li>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><msub><mi mathvariant="normal">O</mi><mtext>vni</mtext></msub><mo>(</mo><mn>2</mn><mo separator="true">;</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{O_{\text{vni}}(2;1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mathrm mtight">vni</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span></span></span></li>
</ul>
<p>Donc :</p>
<ul>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mover accent="true"><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi></mrow><mo>⃗</mo></mover><mo>(</mo><mo>−</mo><mn>4</mn><mo separator="true">;</mo><mn>3</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{\vec{AB}(-4;3)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9663299999999999em;"></span><span class="strut bottom" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">A</span><span class="mord mathrm">B</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">4</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">3</span><span class="mclose">)</span></span></span></span></span></span></li>
<li><span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mover accent="true"><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">O</mi></mrow><mo>⃗</mo></mover><mo>(</mo><mo>−</mo><mn>3</mn><mo separator="true">;</mo><mn>1</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathrm{\vec{AO}(-3;1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9663299999999999em;"></span><span class="strut bottom" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="base"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathrm">A</span><span class="mord mathrm">O</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">3</span><span class="mpunct">;</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span></span></span></li>
</ul>
<p>Alors :</p>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover><mspace width="0.277778em"></mspace><mi mathvariant="normal">.</mi><mspace width="0.277778em"></mspace><mover accent="true"><mrow><mi>A</mi><msub><mi>O</mi><mtext>vni</mtext></msub></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mrow><mo fence="true">(</mo><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>4</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>3</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mi mathvariant="normal">.</mi><mrow><mo fence="true">(</mo><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>3</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>)</mo><mo>=</mo><mo>−</mo><mn>4</mn><mo>∗</mo><mo>(</mo><mo>−</mo><mn>3</mn><mo>)</mo><mo>+</mo><mn>3</mn><mo>∗</mo><mn>1</mn><mo>=</mo><mn>1</mn><mn>5</mn></mrow><annotation encoding="application/x-tex">\overrightarrow{AB}\;.\;\overrightarrow{AO_{\text{vni}}} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}.\begin{pmatrix} -3 \\ 1 \end{pmatrix}) = -4 * (-3) + 3 * 1 = 15</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.45em;"></span><span class="strut bottom" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="base"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span><span class="mspace thickspace"></span><span class="mord">.</span><span class="mspace thickspace"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31750199999999995em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">vni</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
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11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">4</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">3</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">3</span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mord">5</span></span></span></span></span></div>
<p>Puis :</p>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mn>5</mn><mo>∗</mo><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi></mrow><mo>(</mo><mrow><mo fence="true">(</mo><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>4</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>3</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo separator="true">,</mo><mrow><mo fence="true">(</mo><mtable><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>3</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">15 * \mathrm{cos}(\begin{pmatrix} -4 \\ 3 \end{pmatrix},\begin{pmatrix} -3 \\ 1 \end{pmatrix})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.45em;"></span><span class="strut bottom" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="base"><span class="mord">1</span><span class="mord">5</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathrm">c</span><span class="mord mathrm">o</span><span class="mord mathrm">s</span></span><span class="mopen">(</span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">4</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">3</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mclose">)</span></span></span></span></span></div>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mn>5</mn><mo>∗</mo><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi></mrow><mo>(</mo><mo>−</mo><mn>4</mn><mo>∗</mo><mn>1</mn><mo>−</mo><mn>3</mn><mo>∗</mo><mo>(</mo><mo>−</mo><mn>3</mn><mo>)</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">15 * \mathrm{cos}(-4 * 1 - 3 * (-3))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord">1</span><span class="mord">5</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathrm">c</span><span class="mord mathrm">o</span><span class="mord mathrm">s</span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">4</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span></span></div>
<div class="math"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mn>5</mn><mo>∗</mo><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi></mrow><mo>(</mo><mn>5</mn><mo>)</mo><mo>=</mo><mn>1</mn><mn>4</mn><mi mathvariant="normal">.</mi><mn>9</mn><mn>4</mn></mrow><annotation encoding="application/x-tex">15 * \mathrm{cos}(5) = 14.94</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base"><span class="mord">1</span><span class="mord">5</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathrm">c</span><span class="mord mathrm">o</span><span class="mord mathrm">s</span></span><span class="mopen">(</span><span class="mord">5</span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mord">4</span><span class="mord">.</span><span class="mord">9</span><span class="mord">4</span></span></span></span></span></div>Connaître l'angle par rapport au premier point du vecteur, message #1929472018-11-12T14:54:44+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192947<figure><blockquote>
<p> <img src="/static/smileys/huh.png" alt=":o" class="smiley"> Je ne savais pas que c’était la même chose.</p>
</blockquote><figcaption><a href="https://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192943">A-312</a></figcaption></figure>
<p>Ce n’est pas la même chose, mais comme les deux vecteurs sont dans le plan ça revient au même (et c’est plus logique de parler de déterminant car aucun espace n’est défini).</p>
<figure><blockquote>
<p>Alors sur : <a href="http://paquito.amposta.free.fr/glossp/prodscal.htm">http://paquito.amposta.free.fr/glossp/prodscal.htm</a> que signifie "||" ?</p>
</blockquote><figcaption><a href="https://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192943">A-312</a></figcaption></figure>
<p>C’est la norme du vecteur.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929432018-11-12T14:24:39+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192943<blockquote>
<p>le produit vectoriel (ou le déterminant)</p>
</blockquote>
<p> <img src="/static/smileys/huh.png" alt=":o" class="smiley"> Je ne savais pas que c’était la même chose.</p>
<hr>
<p>Alors sur : <a href="http://paquito.amposta.free.fr/glossp/prodscal.htm">http://paquito.amposta.free.fr/glossp/prodscal.htm</a> que signifie "||" ?</p>
<p>EDIT : Mauvais lien + Suppression de la partie, j’allais dans une mauvaise direction.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929402018-11-12T13:54:58+01:00entwanne/@entwannehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192940<p>Salut,</p>
<p>Dans ton autre sujet on évoquait le produit scalaire et le produit vectoriel (ou le déterminant). L’un est fonction des normes des vecteur et du cosinus de l’angle, l’autre des normes et du sinus. Donc tu as tout ce qu’il te faut pour retrouver l’angle.</p>Connaître l'angle par rapport au premier point du vecteur, message #1929392018-11-12T13:51:08+01:00anonyme/@anonymehttps://zestedesavoir.com/forums/sujet/11589/connaitre-langle-par-rapport-au-premier-point-du-vecteur/?page=1#p192939<p>Bonjour,</p>
<p>Grâce à <a href="https://zestedesavoir.com/forums/sujet/11587/savoir-si-le-point-est-a-gauche-ou-a-droite-dun-vecteur/?page=1#p192925">mon autre sujet</a>, je sais si l’OVNI est à ma gauche ou à ma droite. Maintenant je souhaite pouvoir cadrer la photo correctement, je dois donc connaitre l’angle de l’OVNI avec moi (point A<del>-312. <img src="/static/smileys/rire.gif" alt=":lol:" class="smiley"> C’est logique, je suis le point A!</del> ) (donc l’angle <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle><msup><mo><mtext>AOB</mtext></mo><mo>⌢</mo></msup></mstyle></mrow><annotation encoding="application/x-tex">\stackrel \frown \text{AOB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.233678em;"></span><span class="strut bottom" style="height:1.233678em;vertical-align:0em;"></span><span class="base"><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.233678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord text"><span class="mord">AOB</span></span></span></span></span><span style="top:-3.88333em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⌢</span></span></span></span></span></span></span></span></span></span></span></span>).</p>
<p>Comment dois-je procéder pour connaître l’angle <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle><msup><mo><mtext>AOB</mtext></mo><mo>⌢</mo></msup></mstyle></mrow><annotation encoding="application/x-tex">\stackrel \frown \text{AOB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.233678em;"></span><span class="strut bottom" style="height:1.233678em;vertical-align:0em;"></span><span class="base"><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.233678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord text"><span class="mord">AOB</span></span></span></span></span><span style="top:-3.88333em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⌢</span></span></span></span></span></span></span></span></span></span></span></span> ?</p>
<hr>
<h5 id="ma-supposition">Ma supposition :<a aria-hidden="true" href="#ma-supposition"><span class="icon icon-link"></span></a></h5>
<p>Je pense que je dois utiliser Pythagore, une fois que je connais le côté adjacent de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mstyle><msup><mo><mtext>AOB</mtext></mo><mo>⌢</mo></msup></mstyle></mrow><annotation encoding="application/x-tex">\stackrel \frown \text{AOB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.233678em;"></span><span class="strut bottom" style="height:1.233678em;vertical-align:0em;"></span><span class="base"><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.233678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop"><span class="mord text"><span class="mord">AOB</span></span></span></span></span><span style="top:-3.88333em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⌢</span></span></span></span></span></span></span></span></span></span></span></span>.</p>
<p>Il me semble que je dois prolonger <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>O</mi></mrow><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec{AO}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.9663299999999999em;"></span><span class="strut bottom" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="base"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.02778em;">O</span></span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2355em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span></span></span></span></span> et placer le point <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>O</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">O^\prime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:0.751892em;vertical-align:0em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> de sorte à former le triangle rectangle <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><msup><mi>O</mi><mo mathvariant="normal">′</mo></msup><mi>B</mi></mrow><annotation encoding="application/x-tex">AO^{\prime}B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:0.751892em;vertical-align:0em;"></span><span class="base"><span class="mord mathit">A</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathit" style="margin-right:0.05017em;">B</span></span></span></span></span>.</p>
<p>Je ne sais pas comment obtenir la position de <span class="inlineMath"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>O</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">O^\prime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:0.751892em;vertical-align:0em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>.</p>
<p>Bon vol,</p>
<p>A.</p>