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164 lines
7.7 KiB
HTML
164 lines
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HTML
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<article id="content" role="main">
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<h1 class="center">Cross Product</h1>
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<p>A <a href="vectors.html">vector</a> has <b>magnitude</b> (how long it is) and <b>direction</b>:</p>
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<p class="center"><img src="images/vector-mag-dir.svg" alt="vector magnitude and direction" height="137" width="267"></p>
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<p><b>Two vectors</b> <span class="center"> can be <b>multiplied</b> using the "<b>Cross Product</b>" <i>(also see <a href="vectors-dot-product.html">Dot Product</a>)</i></span></p>
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<p class="center"><img src="images/vectors-ab.svg" alt="vectors a and b" height="" width=""></p>
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<p class="center"></p>
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<p>The Cross Product <b>a × b</b> of two vectors is <b>another vector</b> that is at right angles to both:</p>
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<p class="center"><img src="images/cross-product-simple.svg" alt="cross product ">
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<br>
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And it all happens in 3 dimensions!</p>
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<p>The magnitude (length) of the cross product equals the <a href="../area.html">area of a parallelogram</a> with vectors <b>a</b> and <b>b</b> for sides:</p>
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<p class="center"><img src="images/cross-product-area.svg" alt="cross product area" height="234" width="216"></p>
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<p>See how it changes for different angles:</p>
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<script src="../geometry/images/cross-prod.js"></script>
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<script>crossprodMain();</script>
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<p>The cross product (<i style="color:blue">blue</i>) is:</p>
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<ul>
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<li>zero in length when vectors <b>a</b> and <b>b</b> point in the same, or opposite, direction</li>
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<li>reaches maximum length when vectors <b>a</b> and <b>b</b> are at right angles</li>
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</ul>
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<p>And it can point one way or the other!</p>
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<p>So how do we calculate it?</p>
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<h2>Calculating</h2>
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<h4>We can calculate the Cross Product this way:</h4>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cross-product.svg" alt="cross product with angle and unit vector"></p>
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<p class="center larger"><b>a × b</b> = |<b>a</b>| |<b>b</b>| sin(θ) <b>n</b></p>
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<ul>
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<li>|<b>a</b>| is the magnitude (length) of vector <b>a</b></li>
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<li>|<b>b</b>| is the magnitude (length) of vector <b>b</b></li>
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<li>θ is the angle between <b>a</b> and <b>b</b></li>
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<li><b>n</b> is the <a href="vector-unit.html">unit vector</a> at right angles to both <b>a</b> and <b>b</b></li>
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</ul>
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<p>So the <b>length</b> is: the length of <b>a</b> times the length of <b>b</b> times the sine of the angle between <b>a</b> and <b>b</b>,</p>
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<p>Then we multiply by the vector <b>n</b> so it heads in the correct <b>direction</b> (at right angles to both <b>a</b> and <b>b</b>).</p>
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<p> </p>
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<h4>OR we can calculate it this way:</h4>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cross-product-components.svg" alt="cross product components" ></p>
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<p>When <b>a</b> and <b>b</b> start at the origin point (0,0,0), the Cross Product will end at:</p>
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<ul>
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<li><b>c<sub>x</sub> = a<sub>y</sub>b<sub>z</sub> − a<sub>z</sub>b<sub>y</sub></b></li>
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<li><b>c<sub>y</sub> = a<sub>z</sub>b<sub>x</sub> − a<sub>x</sub>b<sub>z</sub></b></li>
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<li><b>c<sub>z</sub> = a<sub>x</sub>b<sub>y</sub> − a<sub>y</sub>b<sub>x</sub></b></li>
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</ul><div style="clear:both"></div>
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<div class="example">
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<h3>Example: The cross product of <b>a</b> = (2,3,4) and <b>b</b> = (5,6,7)</h3>
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<ul>
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<li>c<sub>x</sub> = a<sub>y</sub>b<sub>z</sub> − a<sub>z</sub>b<sub>y</sub> = 3×7 − 4×6 = −3</li>
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<li>c<sub>y</sub> = a<sub>z</sub>b<sub>x</sub> − a<sub>x</sub>b<sub>z</sub> = 4×5 − 2×7 = 6</li>
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<li>c<sub>z</sub> = a<sub>x</sub>b<sub>y</sub> − a<sub>y</sub>b<sub>x</sub> = 2×6 − 3×5 = −3</li>
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</ul>
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<p>Answer: <span class="center"><b>a × b</b> =</span> (−3,6,−3)</p>
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</div>
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<p> </p>
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<p style="float:right; margin: 5px;"><img src="images/right-hand-rule.jpg" alt="right hand rule" height="188" width="164"></p>
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<h2>Which Direction?</h2>
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<p>The cross product could point in the completely opposite direction and still be at right angles to the two other vectors, so we have the:</p>
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<p class="center"><span class="large">"Right Hand Rule"</span></p>
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<p>With your right-hand, point your index finger along vector <b>a</b>, and point your middle finger along vector <b>b</b>: the cross product goes in the direction of your thumb.</p>
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<p> </p>
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<h2>Dot Product</h2>
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<p>The Cross Product gives a <b>vector</b> answer, and is sometimes called the <b>vector product</b>.</p>
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<p>But there is also the <a href="vectors-dot-product.html">Dot Product</a> which gives a <b>scalar</b> (ordinary number) answer, and is sometimes called the <b>scalar product</b>.</p>
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<p> </p>
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<div class="fun">
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<div style="float:left; margin: 0 10px 5px 0;"><img src="../images/style/smile.svg" alt="joke" height="48" width="48"></div>
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<p>Question: What do you get when you cross an elephant with a banana?</p>
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<p><span class="center">Answer: |<b>elephant</b>| |<b>banana</b>| sin(θ) <b>n</b></span></p>
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</div>
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<p> </p>
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<div class="questions">3038, 3039, 3905, 3040, 3041, 3906, 3907, 3908, 3042, 3043</div>
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<div class="related">
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<a href="vectors.html">Vectors</a>
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<a href="index.html">Algebra Index</a>
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