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<h1 class="center">Dot 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>Here are two vectors:</p>
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<p class="center"><img src="images/vectors.svg" alt="vectors" height="" width=""></p>
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<p>They can be <b>multiplied</b> using the "<b>Dot Product</b>" (also see <a href="vectors-cross-product.html">Cross Product</a>).</p>
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<h2>Calculating</h2>
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<p>The Dot Product is written using a central dot:</p>
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<p class="center"><span class="large"><b>a</b> · <b>b</b></span><br>
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This means the Dot Product of <b>a</b> and <b>b</b></p>
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<p>We can calculate the Dot Product of two vectors this way:</p>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="images/dot-product-1.svg" alt="dot product magnitudes and angle" height="" width=""></p>
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<p class="center"><b>a · b</b> = |<b>a</b>| × |<b>b</b>| × cos(θ)</p>
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<p>Where:<br>
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|<b>a</b>| is the magnitude (length) of vector <b>a</b><br>
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|<b>b</b>| is the magnitude (length) of vector <b>b<br>
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</b> θ is the angle between <b>a</b> and <b>b</b></p>
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<p>So we multiply the length of <b>a</b> times the length of <b>b</b>, then multiply by the cosine of the angle between <b>a</b> and <b>b</b></p>
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<p> </p>
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<p>OR we can calculate it this way:</p>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="images/dot-product-2.svg" alt="dot product components"></p>
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<p class="center"><b>a · b</b> = a<sub>x</sub> × b<sub>x</sub> + a<sub>y</sub> × b<sub>y</sub></p>
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<p>So we multiply the x's, multiply the y's, then add.</p>
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<div style="clear:both"></div>
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<p>Both methods work!</p>
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<p>And the result is a <b>number</b> (called a "scalar" to show it is not a vector).</p>
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<div class="example">
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<h3>Example: Calculate the dot product of vectors <b>a</b> and <b>b</b>:</h3>
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<p class="center"><img src="images/dot-product-ex1.gif" alt="dot product example" height="202" width="214"></p>
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<p><b>a · b</b> = |<b>a</b>| × |<b>b</b>| × cos(θ)</p>
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<div class="so"><b>a · b</b> = 10 × 13 × cos(59.5°) </div>
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<div class="so"><b>a · b</b> = 10 × 13 × 0.5075...</div>
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<div class="so"><b>a · b</b> = 65.98... = 66 (rounded)</div>
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<p>OR we can calculate it this way:</p>
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<p><b>a · b</b> = a<sub>x</sub> × b<sub>x</sub> + a<sub>y</sub> × b<sub>y</sub></p>
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<div class="so"><b>a · b</b> = -6 × 5 + 8 × 12</div>
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<div class="so"><b>a · b</b> = -30 + 96</div>
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<div class="so"><b>a · b</b> = 66</div>
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<p>Both methods came up with the same result (after rounding)</p>
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<p>Also note that we used <b>minus 6</b> for a<sub>x</sub> (it is heading in the negative x-direction)</p>
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</div>
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<p>Note: you can use the <a href="vector-calculator.html">Vector Calculator</a>
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to help you.</p>
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<h2>Why cos(θ) ?</h2>
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<p>OK, to multiply two vectors it makes sense to multiply their lengths together <i><b>but only when they point in the same direction</b></i><b>.</b></p>
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<p>So we make one "point in the same direction" as the other by multiplying by cos(θ):</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;"> <img src="images/dot-product-a-cos.svg" alt="dot product |a| cos(theta)" height="131" width="144"></td>
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<td style="text-align:center; width:50px;"> </td>
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<td style="text-align:center;"><img src="images/dot-product-light.svg" alt="dot product shine light" height="245" width="176"></td>
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</tr>
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<tr>
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<td style="text-align:center;">We take the component of <b>a</b><br>
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that lies alongside <b>b</b></td>
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<td style="text-align:center; width:50px;"> </td>
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<td style="text-align:center;">Like shining a light to see<br>
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where the shadow lies</td>
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</tr>
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</tbody></table>
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<p>THEN we multiply !</p>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td>
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<p>It works exactly the same if we "projected" <b>b</b> alongside <b>a</b> then multiplied:</p>
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<p>Because it doesn't matter which order we do the multiplication:</p>
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<p class="center">|<b>a</b>| × <span class="hilite">|<b>b</b>| × cos(θ)</span> = <span class="hilite">|<b>a</b>| × cos(θ)</span> × |<b>b</b>|</p>
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</td>
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<td><img src="images/dot-product-b-cos.gif" alt="dot product |b| cos(theta)" height="136" width="183"></td>
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</tr>
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</tbody></table>
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<h2>Right Angles</h2>
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<p>When two vectors are at right angles to each other the dot product is <b>zero</b>.</p>
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<div class="example">
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<h3>Example: calculate the Dot Product for:</h3>
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<p class="center"><img src="images/dot-product-right-angle.gif" alt="dot product right angle" height="104" width="164"></p>
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<p><b>a · b</b> = |<b>a</b>| × |<b>b</b>| × cos(θ)</p>
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<div class="so"><b>a · b</b> = |<b>a</b>| × |<b>b</b>| × cos(90°) </div>
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<div class="so"><b>a · b</b> = |<b>a</b>| × |<b>b</b>| × 0 </div>
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<div class="so"><b>a · b</b> = 0</div>
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<p>or we can calculate it this way:</p>
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<p><b>a · b</b> = a<sub>x</sub> × b<sub>x</sub> + a<sub>y</sub> × b<sub>y</sub></p>
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<div class="so"><b>a · b</b> = -12 × 12 + 16 × 9</div>
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<div class="so"><b>a · b</b> = -144 + 144</div>
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<div class="so"><b>a · b</b> = 0</div>
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</div>
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<p>This can be a handy way to find out if two vectors are at right angles.</p>
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<h2>Three or More Dimensions</h2>
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<p>This all works fine in 3 (or more) dimensions, too.</p>
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<p>And can actually be very useful!</p>
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<div class="example">
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<h3>Example: Sam has measured the end-points of two poles, and wants to know <b>the angle between them</b>:</h3>
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<p class="center"><img src="images/dot-product-ex2.gif" alt="dot product 3d" height="261" width="317"></p>
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<p>We have 3 dimensions, so don't forget the z-components:</p>
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<p><b>a · b</b> = a<sub>x</sub> × b<sub>x</sub> + a<sub>y</sub> × b<sub>y</sub> + a<sub>z</sub> × b<sub>z</sub></p>
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<div class="so"><b>a · b</b> = 9 × 4 + 2 × 8 + 7 × 10</div>
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<div class="so"><b>a · b</b> = 36 + 16 + 70 </div>
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<div class="so"><b>a · b</b> = 122</div>
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<p> </p>
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<p>Now for the other formula:</p>
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<p><b>a · b</b> = |<b>a</b>| × |<b>b</b>| × cos(θ)</p>
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<p>But what is |<b>a</b>| ? It is the magnitude, or length, of the vector <b>a</b>. We can use <a href="../pythagoras.html">Pythagoras</a>:</p>
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<ul>
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<li>|<b>a</b>| = √(4<sup>2</sup> + 8<sup>2</sup> + 10<sup>2</sup>)</li>
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<li>|<b>a</b>| = √(16 + 64 + 100)</li>
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<li>|<b>a</b>| = √180</li>
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</ul>
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<p>Likewise for |<b>b</b>|:</p>
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<ul>
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<li>|<b>b</b>| = √(9<sup>2</sup> + 2<sup>2</sup> + 7<sup>2</sup>)</li>
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<li>|<b>b</b>| = √(81 + 4 + 49)</li>
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<li>|<b>b</b>| = √134</li>
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</ul>
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<p>And we know from the calculation above that <b>a · b</b> = 122, so:</p>
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<p><b>a · b</b> = |<b>a</b>| × |<b>b</b>| × cos(θ)</p>
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<div class="so">122 = √180 × √134 × cos(θ) </div>
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<div class="so">cos(θ) = 122 / (√180 × √134)</div>
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<div class="so">cos(θ) = 0.7855...</div>
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<div class="so">θ = cos<sup>-1</sup>(0.7855...) = 38.2...°</div>
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<p>Done!</p>
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</div>
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<p>I tried a calculation like that once, but worked all in angles and distances ... it was very hard, involved lots of trigonometry, and my brain hurt. The method above is much easier.</p>
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<h2>Cross Product</h2>
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<p>The Dot Product gives a <b>scalar</b> (ordinary number) answer, and is sometimes called the <b>scalar product</b>.</p>
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<p>But there is also the <a href="vectors-cross-product.html">Cross Product</a> which gives a <b>vector</b> as an answer, and is sometimes called the <b>vector product</b>.</p>
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<p> </p>
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<div class="questions">3036, 3037, 3030, 3031, 3032, 3033, 3034, 3035, 3903, 3904</div>
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<div class="related">
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<a href="vectors.html">Vectors</a>
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<a href="vector-calculator.html">Vector Calculator</a>
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<a href="index.html">Algebra Index</a>
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