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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">Vector Product</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/vectors/vector_product/image-1.png" style="float: left; width: 100px; height: 91px;"></p> <p>The vector product is a way of combining two vectors to find a third vector that is <strong>perpendicular</strong> to the first two. The vector product is a <strong>vector</strong>! Don't mix it up with the <a href="../652/vectors-and-angles.html" title="Vectors and Angles">scalar product</a> which is a <strong>scalar! </strong>The vector product is useful for finding the <a href="../650/equations-of-planes.html" title="Equations of Planes">equation of a plane</a>. The properties are important, so learn them. The two following formula give the definition of the vector product</p> <p style="text-align: center;"><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)</span></p> <p style="text-align: center;"><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left| \textbf{v} \right| \left| \textbf{w} \right| \sin\theta \ \textbf{n}\)</span></p> <p style="text-align: right;">where <span class="math-tex">\(\theta\)</span> is the angle between <em><strong>v</strong></em> and <em><strong>w</strong></em> and <em><strong>n</strong></em> is the unit normal vector</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"> <p>On this page, you should learn about</p> <ul> <li>the vector product (or cross product) of two vectors</li> <li>properties of the vector product</li> <li>the magnitude of the vector product to find the area of a parallelogram</li> </ul> </div> <div class="panel-footer"> <div> </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 from 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>Vector Product - How to</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="358"> <div> <p>In the following video we see how to work out the vector product using the formula, what this means and how to use the right-hand rule to find its direction.</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/252764846"></iframe></div> </div> <div class="smart-object center" data-id="333"> <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/vector_product/vector-product.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/vector-product.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </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>Properties of Vector Product</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="357"> <div> <p>There are 4 important properties of the vector product that you will need to learn. These properties are often used in proofs.</p> <p><span class="math-tex">\(\textbf{v}\times\textbf{w} =-\textbf{w} \times \textbf{v}\\ \textbf{u} \times (\textbf{v}+\textbf{w} )= \textbf{u} \times \textbf{v}+\textbf{u} \times \textbf{w} \\ (k\textbf{v})\times \textbf{w} =k(\textbf{v}\times \textbf{w})\\ \textbf{v}\times \textbf{v}=0 \)</span></p> <hr> <p><br> Here is an example of a proof involving the vector product</p> <hr class="hidden-separator"> <p>Show that <span class="math-tex">\((\textbf{v}+\textbf{w})\times (\textbf{v}-\textbf{w})=2\textbf{w}\times \textbf{v}\)</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/268597576"></iframe></div> </div> <div class="smart-object center" data-id="333"> <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/vector_product/properties-of-vector-product1.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/properties-of-vector-product1.pdf" width="640"></iframe></p> </section> </div> </div> </div> </div> <div class="panel-footer"> <div> </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>Magnitude of Vector Product</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="359"> <p>We know that <span class="math-tex">\(\textbf{v}\times \textbf{w}= \color{red} {\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta} \ \textbf{n}\)</span></p> <p>where <span class="math-tex">\(\theta\)</span> is the angle between <em><strong>v</strong></em> and <em><strong>w</strong></em> and <em><strong>n</strong></em> is the <strong>unit</strong> normal vector.</p> <p>Hence if we find the magnitude of the vector product</p> <p><span class="math-tex">\(\color{red} {\left|\textbf{v}\times \textbf{w}\right|} = \color{red} {\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta} \)</span></p> <p>If we divide this by two, we get something familiar</p> <p><span class="math-tex">\(\frac {1}{2} {\left|\textbf{v}\times \textbf{w}\right|} = \frac {1}{2} {\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta} \)</span></p> <p>You may recall that formula for the area of a triangle is <span class="math-tex">\(A = \frac {1}{2}absin \theta\)</span></p> <p>Hence...</p> <p><img alt="" src="../../files/vectors/vector_product/triangle.jpg" style="float: left; width: 200px; height: 161px;"></p> <p>Area of a triangle = <span class="math-tex">\(\frac {1}{2} {\left|\textbf{v}\times \textbf{w}\right|}\)</span></p> <hr class="hidden-separator"> <p>It therefore follows that for a paralellogram...</p> <p><img alt="" src="../../files/vectors/vector_product/parallelogram.jpg" style="width: 200px; height: 163px; float: left;"></p> <p>Area of a parallelogram = <span class="math-tex">\(\left| \textbf{v}\times \textbf{w} \right| \)</span></p> <hr class="hidden-separator"> <hr> <p>In the following video, we are going to look at this application of the vector product. In this example we use the vector product to find the area of a parallelogram:</p> <hr class="hidden-separator"> <p><em>A parallelogram has two adjacent sides formed by the vectors 2<strong>i</strong> -<strong> j</strong> + 3<strong>k</strong> and a<strong>i</strong> + 4<strong>k</strong>. The area of the parallelogram is <span class="math-tex">\(\sqrt{26}\)</span>. Find the possible values of a.</em></p> <hr class="hidden-separator"> <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/268735468"></iframe></div> <div class="smart-object center" data-id="333"> <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/vector_product/area-of-parallelogram.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/area-of-parallelogram.pdf" width="640"></iframe></p> </section> </div> </div> </div> </div> <div class="panel-footer"> <div> </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>Vector and Scalar Product</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="360"> <p>Here is an example that combines the vector product and the scalar product</p> <hr class="hidden-separator"> <p><em>Prove that <span class="math-tex">\({ \left| \textbf{v}\times \textbf{w} \right| }^{ 2 }={ \left| \textbf{v} \right| }^{ 2 }{ \left| \textbf{w} \right| }^{ 2 }-{ (\textbf{v}\cdot \textbf{w}) }^{ 2 }\)</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/268778959"></iframe></div> <div class="smart-object center" data-id="333"> <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/vector_product/vector-and-scalar-product-proof1.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/vector-and-scalar-product-proof1.pdf" width="640"></iframe></p> </section> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </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/vector_product/vector_product-revision-notes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/vector_product/vector_product-revision-notes.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>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#7f0ec39b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="7f0ec39b"> <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%;"> Vector Product <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="11-185-668" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Find a</p><p><span class="math-tex">\(\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) \times \left( \begin{matrix} 0 \\ -1 \\ 2 \end{matrix} \right) =\left( \begin{matrix} 4 \\ -2 \\ a \end{matrix} \right) \)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p><span></span><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)</span><span></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>Find b</p><p><span class="math-tex">\(\left( \begin{matrix} 2 \\ -2 \\ 1 \end{matrix} \right) \times \left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) =\left( \begin{matrix} -2 \\ b \\ 6 \end{matrix} \right) \)</span></p></div><div class="q-answer"><p>b = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)</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 class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(i\times i\quad =\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>The vector product of two parallel vectors = 0</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><img alt="" src="../../files/vectors/vector_product/axes.jpg" style="width: 300px; height: 216px;"></p><p><span class="math-tex">\(i\times j\quad =\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> -<em>k</em></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(j\times i\)</span></label></p><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input class="c" type="radio"> <em>k</em><span class="radio"> </span></label></p></div><div class="q-explanation"><p>Use the right hand rule: first finger pointing in x direction, second finger pointing in y direction.</p><p>Your thumb should be pointing along the z axis.</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><img alt="" src="../../files/vectors/vector_product/axes.jpg" style="width: 300px; height: 216px;"></p><p><span class="math-tex">\(k\times j=\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-j\times k\)</span></label></p><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input type="radio"> <strong><em>i</em></strong></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{v}\times\textbf{w} =-\textbf{w} \times \textbf{v}\)</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><img alt="" src="../../files/vectors/vector_product/axes.jpg" style="width: 300px; height: 216px;"></p><p><span class="math-tex">\(i\times k=\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> -<strong><em>j</em></strong></label></p><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(k\times i\)</span></label></p><p><label class="radio"><input type="radio"> <strong><em>j</em></strong></label></p></div><div class="q-explanation"><p>Use the right hand rule: first finger pointing in x direction, second finger pointing in z direction.</p><p>Your thumb should be pointing along the negative y direction.</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>Find all the vectors which are perpendicular to <span class="math-tex">\(a\times b\)</span></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(b\times a\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <strong><em>b</em></strong></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <strong><em>a</em></strong></label></p><p><label class="checkbox"><input type="checkbox"> none of the answers</label></p></div><div class="q-explanation"><p><span class="math-tex">\(a\times b\)</span> is a vector perpendicular to <strong><em>a</em></strong> and <strong><em>b</em></strong></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>Which of the following is equal to <span class="math-tex">\(\left| \textbf{v}\times \textbf{w}\right|\)</span></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> area of parallelogram</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\left| \textbf{v} \right| \left| \textbf{w} \right| sin\theta \textbf{n}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\left| \textbf{v} \right| \left| \textbf{w} \right| cos\theta\)</span></label></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><span class="math-tex">\((a-b)\times (a+b)=\quad \)</span></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="2a"> <span class="review"></span><span class="math-tex">\(\times b\)</span></p></div><div class="q-explanation"><p><span class="math-tex">\((a-b)\times (a+b)=a\times a+a\times b-b\times a-b\times b\\=0 +a\times b+a\times b+0\\=2a\times b \)</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">\(a\cdot (a\times b)=\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left| a \right| ²\ \times \ a\cdot b\quad \)</span></label><label class="radio"></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(a\cdot a\ \times \ a\cdot b\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(a\times b\)</span></label></p><p><label class="radio"><input class="c" type="radio"> 0</label></p></div><div class="q-explanation"><p><span class="math-tex">\(a\times b\)</span> is perpendicular to <em>a.</em></p><p>The scalar product of two perpendicular vector = 0</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> 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 <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> <div class="smart-object center" data-id="362"> <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><em><strong>a</strong> </em>= <span class="math-tex">\(\left( \begin{matrix} 2 \\ 3 \\ -5 \end{matrix} \right) \)</span> and <strong><em>b </em></strong>= <span class="math-tex">\(\left( \begin{matrix} 3 \\ -2 \\ 4 \end{matrix} \right) \)</span></p> <p>Find <span class="math-tex">\(\textbf{a}\times \textbf{b}\)</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><span></span><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)</span><span></span></p> <p><strong><em>a </em></strong>x <strong><em>b </em></strong>is perpendicular to <strong><em>a</em></strong> and <strong><em>b</em></strong>. Check your answer by showing <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{a}=0\)</span> and <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{b}=0\)</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/vector_product/esq_vector_vproduct1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </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="363"> <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><em><strong>a</strong> </em>= 3<strong><em>i</em></strong> + 2<strong><em>j</em></strong> + <strong><em>k</em></strong> and <strong><em>b </em></strong>= 2<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 2<strong><em>k</em></strong></p> <p>Find a unit vector that is perpendicular to <strong><em>a</em></strong> and <strong><em>b</em></strong></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><span></span>Use the vector product</p> <p><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)</span></p> <p><span></span></p> <p>A unit vector is a vector with magnitude = 1</p> <p><strong><em>a </em></strong>x <strong><em>b </em></strong>is perpendicular to <strong><em>a</em></strong> and <strong><em>b</em></strong>. Check your answer by showing <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{a}=0\)</span> and <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{b}=0\)</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/vector_product/esq_vector_vproduct2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </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="364"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The area of a parallelogram formed by two adjacent vectors <strong><em>a </em></strong>and <strong><em>b</em></strong> is 7 square units.</p> <p><strong><em>a </em></strong>= <span class="math-tex">\(\left( \begin{matrix} -3 \\ 4 \\ k \end{matrix} \right) \)</span> <strong><em>b</em></strong> = <span class="math-tex">\(\left( \begin{matrix} 3 \\ -2 \\ -2 \end{matrix} \right) \)</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><span>Area of parallelogram = </span> <span></span><span class="math-tex">\(\left| \textbf{a}\times \textbf{b} \right| \)</span><span></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/vector_product/esq_vector_vproduct3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </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="365"> <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>Given that <span class="math-tex">\(\textbf{u}\times \textbf{v} = \textbf{u}\times \textbf{w}\)</span> show that <span class="math-tex">\(\textbf{v}- \textbf{w} \)</span> is parallel to <span class="math-tex">\(\textbf{u}\)</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><span>If <em><strong><span class="math-tex">\(\textbf{a}\times \textbf{b}=0\)</span> </strong></em>then <strong><em>a</em></strong> and <strong><em>b</em></strong> are parallel</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> <span class="math-tex">\(\textbf{u}\times \textbf{v} = \textbf{u}\times \textbf{w}\)</span></p> <p><span class="math-tex">\(\textbf{u}\times \textbf{v} - \textbf{u}\times \textbf{w}=0\)</span></p> <p><span class="math-tex">\(\textbf{u}\times ( \textbf{v} - \textbf{w})=0\)</span></p> <p>Since the vector product is zero, the vectors are parallel</p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </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="366"> <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>a) For any two vectors <strong><em>v </em></strong> and <strong><em>w</em></strong> prove Lagrange's Identity</p> <p style="text-align: center;"><span class="math-tex">\({ \left| \textbf{v}\times \textbf{w} \right| }^{ 2 }+{ \left( \textbf{v}\cdot \textbf{w} \right) }^{ 2 }={ \left| \textbf{v} \right| }^{ 2 }{ \left| \textbf{w} \right| }^{ 2 }\)</span></p> <p>b) <strong>Hence</strong>, find <span class="math-tex">\(\textbf{v}\cdot \textbf{w}\)</span> if</p> <p style="text-align: center;"><span class="math-tex">\({ \left| \textbf{v} \right| }=3\)</span></p> <p style="text-align: center;"><span class="math-tex">\({ \left| \textbf{w} \right| }=4\)</span></p> <p style="text-align: center;"><span class="math-tex">\(\textbf{v}\times \textbf{w} =\left( \begin{matrix} -1 \\ 2 \\ 3 \end{matrix} \right) \)</span><strong> </strong></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>Look in formula booklet for formulae for <span class="math-tex">\({ \left| \textbf{v}\times \textbf{w} \right| }\)</span> and <span class="math-tex">\(\textbf{v}\cdot \textbf{w}\)</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/vector_product/esq_vector_vproduct6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct6.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </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 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="367"> <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>The points A, B and C are given by the position vectors <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</em></strong>.</p> <p>If A, B and C are collinear, show that</p> <p style="text-align: center;"><span class="math-tex">\(\textbf{b}\times \textbf{c}=\textbf{a}\times (\textbf{c}-\textbf{b})\)</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>If A, B and C are collinear, then <strong><em><span class="math-tex">\(\overrightarrow { AB } \)</span></em></strong><strong><em> </em></strong> is parallel to <strong><em><span class="math-tex">\(\overrightarrow { BC } \)</span></em></strong></p> <p><img alt="" src="../../files/vectors/vector_product/esqcollinear1.jpg" style="width: 400px; height: 309px;"></p> <p>What do you know about the vector product of parallel vectors?</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/vector_product/esq_vector_vproduct7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct7.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </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 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="452"> <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><strong><em>a</em></strong> and <strong><em>b</em></strong> are vectors</p> <p>Show that <span class="math-tex">\(|\textbf{a}×\textbf{b}|^{ 2 }+|\textbf{a} ∙\textbf{b}|^{ 2 }=(\textbf{a}\textbf{b})^{ 2 }\)</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"><span class="math-tex">\(|\textbf{a}×\textbf{b}|=\textbf{a}\textbf{b}sin\theta \)</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/vector_product/esq_vector_vproduct8.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct8.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="668"> <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>Vector Product</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My 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