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title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Scalar Product and Angles </span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30&apos;</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/vectors/vectors_angles/main.jpg" style="float: left; width: 100px; height: 100px;"> In this topic, we will look at finding the angle between vectors in different circumstances. The whole topic revolves around the scalar (or dot) product. Often we are concerned with perpendicular vectors, and the fact that the scalar product equals zero in this case is a hugely important result. Since problems are often set in 3 dimensions, the ability to visualise the situations or draw a quick sketch can help.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>the scalar product of two vectors</li> <li>the angle between two vectors</li> <li>parallel and perpendicular vectors</li> </ul> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following videos will help you understand all the concepts&nbsp;from&nbsp;this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Scalar Product &amp; Angles</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="330"> <p>In the following video, we are going to look at the scalar product and how we can use it to find the angle between two vectors. The scalar product, or the dot product as it is sometimes called, is a way of combining two vectors together that will give us a result that is a <strong>scalar</strong> (not a <strong>vector</strong>). The formula for finding the scalar product for 3D can be found in the formula booklet (it is not there for 2D)</p> <table align="center" border="0" cellpadding="0" cellspacing="0" style="width: 70%;"> <tbody> <tr> <td style="text-align: center; vertical-align: middle;"> <p>2 dimensions</p> </td> <td style="text-align: center; vertical-align: middle;"><span class="math-tex">\({ v }\cdot { w }=\left( \begin{matrix} { v }_{ 1 } \\ { v }_{ 2 } \end{matrix} \right) \cdot \left( \begin{matrix} { w }_{ 1 } \\ { w }_{ 2 } \end{matrix} \right) ={ v }_{ 1 }\cdot { w }_{ 1 }+{ v }_{ 2 }\cdot { w }_{ 2 }\)</span></td> </tr> <tr> <td style="text-align: center; vertical-align: middle;"> <p>3 dimensions</p> </td> <td style="text-align: center; vertical-align: middle;"><span class="math-tex">\(\textbf{ v}\cdot \textbf{w }=\left( \begin{matrix} { v }_{ 1 } \\ { v }_{ 2 } \\ { v }_{ 3 } \end{matrix} \right) \cdot \left( \begin{matrix} { w }_{ 1 } \\ { w }_{ 2 } \\ { w }_{ 3 } \end{matrix} \right) ={ v }_{ 1 }\cdot { w }_{ 1 }+{ v }_{ 2 }\cdot { w }_{ 2 }+{ v }_{ 3 }\cdot { w }_{ 3 }\)</span></td> </tr> </tbody> </table> <p>We&#39;ll look at the following example</p> <p><em>Find the angle between two direction vectors <span class="math-tex">\(\left( \begin{matrix} 1 \\ 2 \\ \sqrt{3} \end{matrix} \right) \)</span> and <span class="math-tex">\(\left( \begin{matrix} -1 \\ 3 \\-2 \end{matrix} \right) \)</span></em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/252931804"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/scalar-product-and-angle-between-vectors.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/scalar-product-and-angle-between-vectors.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Perpendicular Vectors</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="331"> <p>When two vectors <strong>v </strong>and <strong>w</strong> are perpendicular vectors, then <span class="math-tex">\(\textbf{v}\cdot \textbf{w}=0\)</span></p> <hr class="hidden-separator"> <p>In the following video, we look at the following example</p> <p><em>Find <strong>a</strong> if the following two vectors are perpendicular</em></p> <p><span class="math-tex">\(2\textbf{i}−4\textbf{j}+a\textbf{k}\\ a\textbf{i}+\sqrt{3} \textbf{j}−\textbf{k}\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/253005499"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/perpendicular-vectors.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/perpendicular-vectors.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Angles between 2 Lines</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="332"> <p>In the following video we are looking at finding the angle between two lines in 3 dimensional space. The formula for finding the angle between 2 vectors <strong><em>a</em> </strong>and <strong><em>b</em> </strong>is given in the formula booklet</p> <p style="text-align: center;"><span class="math-tex">\(cos\theta =\frac { a\cdot b }{ \left| a \right| \left| b \right| } \)</span></p> <p>The example we are going to try and solve is</p> <p><em>Find the angle between the lines</em></p> <p style="text-align: center;"><em><span class="math-tex">\({ L }_{ 1 }:\quad \frac { x-1 }{ -1 } =-y=\frac { z+2 }{ \sqrt { 3 } } \\ { L }_{ 2 }:\quad \frac { x+2 }{ -2 } =\frac { 2y+1 }{ -4 } =z+1\)</span></em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/252235745"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/angle-between-2-lines.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/angle-between-2-lines.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Angle between 2 Planes</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="333"> <p>In the following video we are go to find out how we find the angle between two planes. It is easy to find the angle when the planes are given in Cartesian form as the angle required is the angle between the normals</p> <p style="text-align: center;"><img alt="" src="../../files/vectors/planes/angle-between-planes.jpg" style="width: 300px; height: 240px;"></p> <p>Whenever we dealing with angles, the formula (found in formula booklet) involving the scalar product is going to be useful</p> <p style="text-align: center;"><span></span><span class="math-tex">\(cos\theta =\frac { a\cdot b }{ \left| a \right| \left| b \right| } \)</span><span></span></p> <p>Here is the example that will help us understand this topic:</p> <p><em>Find the acute angle between the planes</em></p> <p><em>2x + 3y &ndash; 4z = 6</em></p> <p><em>and </em></p> <p><em>x - y + 2z = 2</em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/251691304"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/angle-between-2-planes.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/angle-between-2-planes.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Angle between a Line and a Plane</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="334"> <div>In the following video we are going to see how we can find the angle between a line and a plane. The key thing is to be able to visualise the problem and to understand what you need to work out. The diagram below should help you see that the angle we need is <span class="math-tex">\(\alpha\)</span> but the angle we can easily find using the normal to the plane is <span class="math-tex">\(\theta\)</span>. The video below explains this in detail. <p><img alt="" src="../../files/vectors/planes/angleplaneandline.jpg" style="width: 400px; height: 382px;"></p> <p>As ever, when we are dealing with angles, the following formula (in the formula booklet) is going to be useful</p> <p style="text-align: center;"><span></span><span class="math-tex">\(cos\theta =\frac { a\cdot b }{ \left| a \right| \left| b \right| } \)</span><span></span></p> <p>The example we are going to look at is as follows:</p> <p><em>Find the acute angle between the line and plane</em></p> <p><span class="math-tex">\(-x\ =\ \frac { y-5 }{ 2 } \ =\ 2z-8\)</span></p> <p><em>and </em></p> <p><em>3x - y + z = 8</em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/251855841"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/angle-between-a-line-and-plane.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/angle-between-a-line-and-plane.pdf" width="640"></iframe></p> </section> </div> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/revision-notes_vectors_angleshl1.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/vectors_angles/revision-notes_vectors_angleshl1.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>The following quiz tests your knowledge and understanding of the scalar product, angles between vectors and angles between lines</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#10683dfb"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="10683dfb"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Vectors and Angles 1 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-181-652" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><em><strong>v</strong></em> = <span class="math-tex">\(\left( \begin{matrix} 2 \\ -3 \end{matrix} \right) \)</span> and <strong><em>w </em></strong>= <span class="math-tex">\(\left( \begin{matrix} -1 \\ 4 \end{matrix} \right) \)</span></p><p>Which of the following is correct about the scalar product <span class="math-tex">\(\textbf{ v }\cdot \textbf{ w }\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\textbf{ w }\cdot \textbf{ v }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\textbf{ v }\cdot \textbf{ w }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\({ 2 }\cdot (-1)-{ (-3) }\cdot 4\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\({ 2 }\cdot (-1)\times { (-3) }\cdot 4\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{ v }\cdot \textbf{ w }=\textbf{ w }\cdot \textbf{ v }={ 2 }\cdot (-1)+ { (-3) }\cdot 4\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>For the vectors<strong><em> v </em></strong>= -<strong><em>j</em></strong> +2<strong><em>k</em></strong> and <strong><em>w</em></strong> = <strong><em>i</em></strong> - 2<strong><em>j</em></strong> + 3<strong><em>k</em></strong> , find <span class="math-tex">\(\textbf{ v }\cdot \textbf{ w }\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\textbf{ v }\cdot \textbf{ w }\)</span>= <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>It is usually easier to write in vector notation <span class="math-tex">\(\left( \begin{matrix} 0 \\ -1 \\ 2 \end{matrix} \right) \cdot \left( \begin{matrix} 1 \\ -2 \\ 3 \end{matrix} \right) =0\cdot 1+(-1)\cdot (-2)+2\cdot 3=8\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>For the vector <em><strong>a</strong></em> = 6<strong><em>i</em></strong> - 3<strong><em>k</em></strong> + 2<strong><em>j</em></strong> , find <span class="math-tex">\(\left| \textbf{a} \right| \)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\left| \textbf{a} \right| \)</span> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\sqrt { { 6 }^{ 2 }+{ (-3) }^{ 2 }+{ 2 }^{ 2 } } =\sqrt { 36+9+4 } =\sqrt { 49 } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\theta\)</span> is the angle between the vectors <strong><em>a</em></strong> and <strong><em>b</em></strong> such that</p><p><span class="math-tex">\(\textbf{ a}\cdot \textbf{ b }=-6\)</span></p><p><span class="math-tex">\(\left| \textbf{a} \right| =3\)</span></p><p><span class="math-tex">\(\left| \textbf{b} \right| =4\)</span></p><p>Find <span class="math-tex">\(cos\theta\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(cos\theta\)</span> = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(cos\theta =\frac {\textbf {a}\cdot \textbf{b} }{ \left| \textbf{a} \right| \left| \textbf{b} \right| } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><strong><em>v</em></strong> and <strong><em>w </em></strong> are perpendicular vectors. Find <span class="math-tex">\(\textbf{ v }\cdot \textbf{ w }\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\textbf{ v }\cdot \textbf{ w }\)</span> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><em><strong>v</strong></em> = <span class="math-tex">\(\left( \begin{matrix} 2 \\ -1 \end{matrix} \right) \)</span> and <strong><em>w</em></strong> = <span class="math-tex">\(\left( \begin{matrix} a\\ 4 \end{matrix} \right) \)</span>.</p><p><strong><em>v</em></strong> and <strong><em>w</em></strong> are perpendicular vectors. Find <strong>a</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 2 \\ -1 \end{matrix} \right) \cdot \left( \begin{matrix} a \\ 4 \end{matrix} \right) =0\\ 2a\quad -4\quad =\quad 0\\ a=2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\theta\)</span> is the angle between the lines L<sub>1</sub> and L<sub>2</sub> .</p><p><span class="math-tex">\({ L }_{ 1 }:\quad \textbf{r}=\left( \begin{matrix} 0 \\ 2 \end{matrix} \right) +\lambda \left( \begin{matrix} -3 \\ 4 \end{matrix} \right) \)</span></p><p><span class="math-tex">\({ L }_{ 2 }:\quad \textbf{r}=\left( \begin{matrix} -1 \\ 5 \end{matrix} \right) +\mu \left( \begin{matrix} 2\\ 1 \end{matrix} \right) \)</span></p><p> <span class="math-tex">\(cos\theta=\frac { a }{ b\sqrt { 5 } } \)</span>. Find a and b</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} -3 \\ 4 \end{matrix} \right) \cdot \left( \begin{matrix} 2 \\ 1 \end{matrix} \right) =-2\)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} -3 \\ 4 \end{matrix} \right) \right| =5\)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} 2 \\ 1 \end{matrix} \right) \right| =\sqrt { 5 } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>L<sub>1</sub> is a line that passes through the point (5,-6,7) and is parallel to 2<strong><em>i</em></strong></p><p>L<sub>2</sub> : <span class="math-tex">\(\textbf{r}=2\textbf{i}-\textbf{k}+\lambda (3\textbf{i}-\textbf{j}+2\textbf{k})\)</span></p><p><span class="math-tex">\(\theta\)</span> is the angle between the lines L<sub>1</sub> and L<sub>2</sub> .</p><p><span class="math-tex">\(cos\theta=\frac { a }{ \sqrt { b } } \)</span></p><p>Find <strong>a</strong> and <strong>b</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="14"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 2 \\ 0 \\ 0 \end{matrix} \right) \cdot \left( \begin{matrix} 3 \\ -1 \\ 2 \end{matrix} \right) =6\)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} 2 \\ 0 \\ 0 \end{matrix} \right) \right| =2\)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} 3 \\ -1 \\ 2 \end{matrix} \right) \right| =\sqrt { 14 } \)</span></p><p><span class="math-tex">\(cos\theta=\frac { 6 }{ 2\sqrt { 14 } } =\frac { 3 }{ \sqrt { 14 } }\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i>&nbsp;&nbsp;Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next&nbsp;&nbsp;<i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>The following quiz tests your knowledge and understanding of angles between</p> <ol> <li>Two Lines</li> <li>Two Planes</li> <li>A Line and a Plane</li> </ol> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d16081d7"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d16081d7"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Vectors and Angles 2 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-182-652" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\Pi_1\)</span> and <span class="math-tex">\(\Pi_2\)</span> are perpendicular planes.</p><p><span class="math-tex">\(\Pi_1\)</span>: 2x - y + z = -3</p><p><span class="math-tex">\(\Pi_2\)</span>: 3x + <strong>a</strong>y - z = 4</p><p>Find <strong>a</strong>.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The normals to the planes are <span class="math-tex">\(\left( \begin{matrix} 2 \\ -1\\ 4 \end{matrix} \right) \)</span> and <span class="math-tex">\(\left( \begin{matrix} 3 \\ a \\ -1 \end{matrix} \right) \)</span></p><p>Since the planes are perpendicular <span class="math-tex">\(\left( \begin{matrix} 2 \\ -1 \\ 4 \end{matrix} \right) \cdot \left( \begin{matrix} 3 \\ a \\ -1 \end{matrix} \right) =0\)</span></p><p>6 - a - 4 = 0</p><p>2 - a = 0</p><p>a = 2</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The angle between the planes <span class="math-tex">\(\Pi_1\)</span> and <span class="math-tex">\(\Pi_2\)</span> is <span class="math-tex">\(\theta\)</span></p><p><span class="math-tex">\(\Pi_1\)</span> : x + 2y + z = 1</p><p><span class="math-tex">\(\Pi_2\)</span> : 2x - y + 3z = 10</p><p><span class="math-tex">\(cos\theta = \frac { \sqrt{a} }{ 6 } \)</span></p><p>Find a</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 1 \\ 2 \\ 1 \end{matrix} \right) \cdot \left( \begin{matrix} 2 \\ -1 \\ 5 \end{matrix} \right) =5\)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} 1 \\ 2 \\ 1 \end{matrix} \right) \right| =\sqrt { 6 } \)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} 2 \\ -1 \\ 5 \end{matrix} \right) \right| =\sqrt { 30 } \)</span></p><p><span class="math-tex">\(cos\theta =\frac { 5 }{ \sqrt { 180} } = \frac { 5 }{ 6\sqrt { 5} }= \frac { \sqrt{5} }{ 6 }\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\theta\)</span> is the angle between the lines L<sub>1</sub> and L<sub>2</sub></p><p><span class="math-tex">\({ L }_{ 1 }:\quad x-2=\frac { y+2 }{ 2 } =-z\)</span></p><p><span class="math-tex">\({ L }_{ 2 }:\quad \frac { x+1 }{ -2 } =y=\frac { z-1 }{ 3 } \)</span></p><p><span class="math-tex">\(cos\theta =\frac { -\sqrt { a } }{ 14 } \)</span></p><p>Find <strong>a</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="21"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\({ L }_{ 1 }:\quad \frac { x-2 }{ 1 } =\frac { y+2 }{ 2 } =\frac { z-0 }{ -1 } \\ { L }_{ 2 }:\quad \frac { x+1 }{ -2 } =\frac { y-0 }{ 1 } =\frac { z-1 }{ 3 } \)</span></p><p><span class="math-tex">\(\left( \begin{matrix} 1 \\ 2 \\ -1 \end{matrix} \right) \cdot \left( \begin{matrix} -2 \\ 1 \\ 3 \end{matrix} \right) =-3\)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} 1 \\ 2 \\ -1 \end{matrix} \right) \right| =\sqrt { 6 } \)</span></p><p><span class="math-tex">\(\left| \left( \begin{matrix} -2 \\ 1 \\ 3 \end{matrix} \right) \right| =\sqrt { 14 }
\)</span></p><p><span class="math-tex">\(cos\theta =\frac { -3 }{ \sqrt { 6 } \sqrt { 14 } } =\frac { -\sqrt { 21 } }{ 14 } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Line L and plane п are parallel:</p><p>L: 2<strong><em>i</em></strong> + 3<strong><em>j</em></strong> -<strong><em> k</em></strong> +&lambda;(<strong><em>i</em></strong> - 2<strong><em>j</em></strong> - 5<strong><em>k</em></strong>)</p><p>п: <strong>k</strong>x + 2y + 3z = 8</p><p>Find <strong>k</strong>.</p></div><div class="q-answer"><p>k = <input type="text" style="height: auto;" data-c="-11"> <span class="review"></span></p></div><div class="q-explanation"><p>If line and plane are parallel then normal to plane and line are perpendicular.</p><p><span class="math-tex">\(\left( \begin{matrix} 1 \\ -2 \\ 5 \end{matrix} \right) \cdot \left( \begin{matrix} k \\ 2 \\ 3 \end{matrix} \right) =0\)</span></p><p>k - 4 + 15 = 0</p><p>k = -11</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Line L and plane п are perpendicular</p><p><span class="math-tex">\(L:\quad \textbf{ r}=\left( \begin{matrix} 3 \\ -3 \\ 2 \end{matrix} \right) +\lambda \left( \begin{matrix} -4 \\ 0 \\ a \end{matrix} \right) \)</span></p><p>п : 2x - 5z = 3</p><p>Find <strong>a</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>If the line and the plane are perpendicular then line and normal are parallel.</p><p><span class="math-tex">\(\left( \begin{matrix} -4 \\ 0 \\ a \end{matrix} \right) =k\left( \begin{matrix} 2 \\ 0 \\ -5 \end{matrix} \right) \)</span></p><p><span class="math-tex">\(\left( \begin{matrix} -4 \\ 0 \\ 10 \end{matrix} \right) =-2\left( \begin{matrix} 2 \\ 0 \\ -5 \end{matrix} \right) \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i>&nbsp;&nbsp;Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next&nbsp;&nbsp;<i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="337"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A line <span class="math-tex">\({ L }_{ 1 }\)</span> passes through A(2,0,-3) and B(4,3,2).</p> <p>a) Find the equation of the line <span class="math-tex">\({ L }_{ 1 }\)</span></p> <p>A second line <span class="math-tex">\({ L }_{ 2}\)</span> has equation <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2 \\ 3 \\ 5 \end{matrix} \right) +\lambda \left( \begin{matrix} 1 \\ -4 \\ k \end{matrix} \right) \)</span></p> <p>b) Given that <span class="math-tex">\({ L }_{ 1 }\)</span> and <span class="math-tex">\({ L }_{ 2 }\)</span>are perpendicular, find k.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) Line <span class="math-tex">\({ L }_{ 1 }\)</span> is parallel to <span class="math-tex">\(\overrightarrow { AB }\)</span></p> <p>b) If <span class="math-tex">\({ L }_{ 1 }\)</span> and <span class="math-tex">\({ L }_{ 2 }\)</span> are perpendicular, then the scalar product of the direction vectors = 0</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles3.pdf" width="640"></iframe></p> </section> <h4>&nbsp;</h4> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="336"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span class="math-tex">\(\overrightarrow { AB }\)</span> and <span class="math-tex">\(\overrightarrow { AC }\)</span> are two vectors such that <span class="math-tex">\(\overrightarrow { AB } =\left( \begin{matrix} 3 \\ -1 \\ 2 \end{matrix} \right) \)</span> and <span class="math-tex">\(\overrightarrow { AC } =\left( \begin{matrix} 2 \\ 0 \\ 1 \end{matrix} \right) \)</span></p> <p>Find <span class="math-tex">\(\hat { BAC } \)</span> to the nearest degree.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content><img alt="" src="../../files/vectors/vectors_angles/esqs/eqs2.jpg" style="width: 200px; height: 121px;"></p> <p><span class="math-tex">\(cos\theta =\frac { a\cdot b }{ \left| a \right| \left| b \right| } \)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles2.pdf" width="640"></iframe></p> </section> <h4>&nbsp;</h4> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="341"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The angle between the line <span class="math-tex">\({ L }_{ 1 }\)</span> and <span class="math-tex">\({ L }_{ 2 }\)</span> is <span class="math-tex">\(\frac{\pi}{2}\)</span>.</p> <p><span class="math-tex">\( { L }_{ 1 }:\quad \frac { x+2 }{ 3 } =2y+1=\frac { 5-z }{ 2 } \)</span></p> <p><span class="math-tex">\( { L }_{ 2 }: \quad x =\frac { y-2}{ 3} =kz \)</span></p> <p>Find k.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>It is a good idea to write the equations of the lines in the standard form</p> <p><span class="math-tex">\(\frac { x-{ x }_{ 0 } }{ l } =\frac { y-{ y }_{ 0 } }{ m } =\frac { z-{ z }_{ 0 } }{ n } \)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles7.pdf" width="640"></iframe></p> </section> <h4>&nbsp;</h4> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="339"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>Find the angle between the planes <span class="math-tex">\({ \Pi }_{ 1 }\)</span> and <span class="math-tex">\({ \Pi }_{ 2 }\)</span> to the nearest degree.</p> <p><span class="math-tex">\({ \Pi }_{ 1 }: 2x-3y+z=0\)</span></p> <p><span class="math-tex">\({ \Pi }_{ 2 }: x+2y+5z=-4\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>angle between two planes = angle between normals</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles5.pdf" width="640"></iframe></p> </section> <h4>&nbsp;</h4> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="342"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the value of x for which the vectors <span class="math-tex">\(\left( \begin{matrix} sinx \\ \sqrt{3} \\ 0 \end{matrix} \right) \)</span> and <span class="math-tex">\(\left( \begin{matrix} 4cosx \\-1\\ 2 \end{matrix} \right) \)</span>are perpendicular, <span class="math-tex">\(0\le x\le \frac { \pi }{ 2 } \)</span>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>sin2x=2sinxcosx</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles8.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles8.pdf" width="640"></iframe></p> </section> <h4>&nbsp;</h4> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="343"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>OABC is a parallelogram.</p> <p><span class="math-tex">\(\overrightarrow { OA } =\textbf{a}\)</span> <strong><em> </em></strong><span class="math-tex">\(\overrightarrow { OB } =\textbf{b}\)</span> <span class="math-tex">\(\overrightarrow { OC } =\textbf{a}+\textbf{b}\)</span></p> <p>Given that <span class="math-tex">\((\textbf{ a }+\textbf{ b })\cdot (\textbf{ a }-\textbf{ b })=0\)</span> what can you conclude</p> <h4>&nbsp;</h4> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <ul> <li><content> </content>Draw a diagram!</li> <li>The distributive law holds for the scalar product</li> <li><span class="math-tex">\(\textbf{ a }\cdot \textbf{ a }={ \left| \textbf{a} \right| }^{ 2 }\)</span></li> </ul> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles9.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles9.pdf" width="640"></iframe></p> </section> <h4>&nbsp;</h4> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="344"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><img alt="" src="../../files/vectors/vectors_angles/esqs/pythagoras.jpg" style="float: right; width: 200px; height: 227px;"><a aria-labelledby="cke_468_label" href="javascript:void(0)" id="cke_469_uiElement" role="button" style="-moz-user-select: none;" title="OK"><span id="cke_468_label"></span></a>ACB is a right-angled triangle</p> <p><span class="math-tex">\(\overrightarrow { CB } = \textbf {a }\)</span> <span class="math-tex">\(\overrightarrow { AC } = \textbf {b }\)</span></p> <p>a) Write <span class="math-tex">\(\overrightarrow { AB } \)</span> in terms of <strong><em>a </em></strong>and <strong><em>b</em></strong></p> <p>b) Find <span class="math-tex">\(\textbf{ a }\cdot \textbf{ b }\)</span></p> <p>c) Show that <span class="math-tex">\({ \left| \textbf { a+b } \right| }^{ 2 }={ \left| \textbf { a } \right| }^{ 2 }+{ \left| \textbf{ b } \right| }^{ 2 }\)</span> and hence prove Pythagoras&#39; Theorem.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content><span class="math-tex">\(\textbf{ a }\cdot \textbf{ a }={ \left| \textbf{a} \right| }^{ 2 }\)</span></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles10.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles10.pdf" width="640"></iframe></p> </section> <h4>&nbsp;</h4> </div> </div> </div> <div class="panel-footer"> <div>&nbsp;</div> </div> </div> </div> </div> <p><a aria-labelledby="cke_468_label" href="javascript:void(0)" id="cke_469_uiElement" role="button" style="-moz-user-select: none;" title="OK"><span id="cke_468_label"></span></a></p> <div class="page-container panel-self-assessment" data-id="652"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div>  <label class="label-lg">Self-assessment</label><p>How much of <strong>Scalar Product and Angles 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