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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">Equation of a Line</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/lines/main.jpg" style="float: left; width: 100px; height: 100px;">You should already be familiar with the equation of a straight line in Cartesian form in 2 dimensions, y = ax + b. When we move into 3 dimensions, the Cartesian form becomes a little more awkward. Don't worry, vectors are here to help us out! Once you understand the vector equation of a line, it is really useful for solving all sorts of problems with angles and intersections.</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>vector equations of lines in two and three dimensions in the three different forms <ul> <li>vector form</li> <li>parametric form</li> <li>cartesian form</li> </ul> </li> </ul> </div> </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 Equation of a Line</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="320"> <p>In the following video we are going to look at the equation of a straight line in vector form and gain a strong conceptual understanding of what this formula means</p> <p style="text-align: center;"><span class="math-tex">\(\textbf{ r }\ =\ \overrightarrow { OA } +\lambda \textbf{b}\)</span></p> <p>You are probably used to Cartesian form of the equation of a straight line (y = mx + c) and perhaps are wondering why you need <strong>vectors</strong> to describe a straight line. There are number of answers to this</p> <ol> <li>The Cartesian form is a bit messy when used in 3D.</li> <li>Finding intersections and the angles which lines meet is easier in vector form.</li> <li>Describing motion is really helpful using velocity vectors.</li> </ol> <p>Let's start by looking at an example in 2D and then we can move into 3D</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/253851747"></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/lines/vector-equation-of-lines.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/vector-equation-of-lines.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>Converting between the different Forms</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="321"> <p>In the following video we are going to look at the three different forms of the equation of a straight line. In particular, we are going to look at how we can convert from one form to another.</p> <table border="0" cellpadding="0" cellspacing="0" style="width: 100%;"> <tbody> <tr> <td style="text-align: center;"> <p><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 1 \\ -2 \\ 3 \end{matrix} \right) +\lambda \left( \begin{matrix} -1 \\ 3 \\ 4 \end{matrix} \right) \)</span></p> </td> <td>Vector Form</td> </tr> <tr> <td style="text-align: center;"> <p><span class="math-tex">\(x=1-\lambda \\ y=-2+3\lambda \\ z=3+4\lambda \)</span></p> </td> <td>Parametric Form</td> </tr> <tr> <td style="text-align: center;"> <p><span class="math-tex">\(\frac { x-1 }{ -1 } =\frac { y-(-2) }{ 3 } =\frac { z-3 }{ 4 } \)</span></p> </td> <td>Cartesian Form</td> </tr> </tbody> </table> <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/254017542"></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/lines/vector_-parametric-and-cartesian-form-of-straight-line.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/vector_-parametric-and-cartesian-form-of-straight-line.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>Example - Equidistant Points</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="327"> <p>In the following video we are going to look at a lovely application of the equation of the straight line to find points equidistant from another point. We don't necessarily have to do it in this way, but it might help us really understand what the vector equation of a straight line means.</p> <p>Here is the example</p> <p><em>A(3,-1,2) and B(6,-7,-7) lie on a straight line L. C also lies on the straight line L. Find the coordinates of the point C given that <span class="math-tex">\(\left| \overrightarrow { AC } \right| =\left| \overrightarrow { AB } \right| \)</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/265402041"></iframe></div> <h4><span></span><span tabindex="-1"><span class="fa fa-pencil" data-widget="FontAwesome" style="color:rgb(0, 0, 0);"></span><span style="background:rgba(220,220,220,0.5);background-image:url(../../../ckeditor/plugins/widget/images/handle.png)"><img draggable="true" height="15" src="data:image/gif;base64,R0lGODlhAQABAPABAP///wAAACH5BAEKAAAALAAAAAABAAEAAAICRAEAOw==" title="Click and drag to move" width="15"></span></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 tabindex="-1"><span class="fa fa-print" data-widget="FontAwesome" style="color:rgb(0, 0, 0);font-size:14px;"></span><span style="background:rgba(220,220,220,0.5);background-image:url(../../../ckeditor/plugins/widget/images/handle.png)"><img draggable="true" height="15" src="data:image/gif;base64,R0lGODlhAQABAPABAP///wAAACH5BAEKAAAALAAAAAABAAEAAAICRAEAOw==" title="Click and drag to move" width="15"></span></span> Print from <a href="../../files/vectors/lines/equidistant-points.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/equidistant-points.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>Velocity Vectors</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="322"> <p>In the following video we are going to look we will try and gain a conceptual understanding of velocity vectors. One of the key ideas of this topic is to decide if objects collide. It is not enough to consider if their paths cross. We need to think about whether they occupy the <strong>same position</strong> at the <strong>same moment in time</strong>.</p> <hr class="hidden-separator"> <p>To get you started, you might like to play <a href="http://www.geogebra.org/m/SkeytGs5" target="_blank">this game</a> to give you an idea about what is going on. Try to hit the submarine with the torpedo!</p> <hr class="hidden-separator"> <p>Now let's consider the example below:</p> <p><em>A submarine is initially positioned at (0, 5) travels with velocity <span class="math-tex">\(\left( \begin{matrix} 4 \\ -3 \end{matrix} \right) \\ \)</span>ms-1 .</em></p> <p><em>One second later a torpedo is fired from (3, 0) with velocity <span class="math-tex">\(\left( \begin{matrix} 5 \\ 1 \end{matrix} \right) \\ \)</span>ms-1 .</em></p> <p><em>Does the torpedo manage to shoot the submarine?</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/264948030"></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/lines/velocity-vectors.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/velocity-vectors.pdf" width="640"></iframe></p> </section> </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/lines/lines_revision-notes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/lines/lines_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 from about equations of lines in vector form</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#37cb90ac"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="37cb90ac"> <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 - Equation of Line <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="11-180-651" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The equation of the line below is given by the equation <span class="math-tex">\(\textbf{r}=\overrightarrow { OA } +\lambda \overrightarrow { AB } \)</span>.</p><p>To describe the<strong> position of C</strong>, what is a possible value of <span class="math-tex">\(\lambda\)</span></p><p style="text-align: center;"><img alt="" src="../../files/vectors/lines/mcq/q1.jpg" style="width: 300px; height: 150px;"></p><p style="text-align: center;"> </p></div><div class="q-answer"><p><label class="radio"><input type="radio"> -0.8</label></p><p><label class="radio"><input type="radio"> 0.5</label></p><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> 1.5</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>The equation of the line below is given by the equation <span class="math-tex">\(\textbf{r}=\overrightarrow { OA } +\lambda \overrightarrow { AB } \)</span>.</p><p>To describe the<strong> position of C</strong>, what is a possible value of <span class="math-tex">\(\lambda\)</span></p><p style="text-align: center;"><img alt="" src="../../files/vectors/lines/mcq/q2.jpg" style="width: 300px; height: 114px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> -0.8</label><label class="radio"><span class="radio"> <span class="radio"></span></span></label></p><p><label class="radio"><input type="radio"> 1.5</label></p><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> 0.5</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>The equation of the line below is given by the equation <span class="math-tex">\(\textbf{r}=\overrightarrow { OA } +\lambda \overrightarrow { AB } \)</span>.</p><p>To describe the<strong> position of C</strong>, what is a possible value of <span class="math-tex">\(\lambda\)</span></p><p style="text-align: center;"><img alt="" src="../../files/vectors/lines/mcq/q3.jpg" style="width: 300px; height: 114px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0.5</label></p><p><label class="radio"><input type="radio"> 1.5</label></p><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> -0.8</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>Which of the following points lie on the line <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 1 \\ -2 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 4 \end{matrix} \right) \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> (3,2)</label></p><p><label class="radio"><input type="radio"> (3,4)</label></p><p><label class="radio"><input type="radio"> (4,3)</label></p><p><label class="radio"><input class="c" type="radio"> (-5,-10)</label></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 1 \\ -2 \end{matrix} \right) +(-2) \left( \begin{matrix} 3 \\ 4 \end{matrix} \right) = \left( \begin{matrix} -5 \\ -10 \end{matrix} \right)\)</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>What is the value of <strong><em>a </em></strong>so that (-7,-8,4) lies on the line <span class="math-tex">\({ r }=\left( \begin{matrix} a \\ -2 \\ 0 \end{matrix} \right) +\lambda \left( \begin{matrix} 4 \\ 3 \\ -2 \end{matrix} \right) \)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> -1</label></p><p><label class="radio"><input type="radio"> -7</label></p><p><label class="radio"><input type="radio"> -11</label></p><p><label class="radio"><input class="c" type="radio"> 1</label></p></div><div class="q-explanation"><p><span class="math-tex">\({ r }=\left( \begin{matrix} 1 \\ -2 \\ 0 \end{matrix} \right) +(-2)\left( \begin{matrix} 4 \\ 3 \\ -2 \end{matrix} \right) \)</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>What are the values of <strong><em>a </em></strong>and <strong><em>b </em></strong>so that (-2,-7,4) lies on the line <span class="math-tex">\({ r }=\left( \begin{matrix} 1 \\ -1 \\ -2 \end{matrix} \right) +\lambda \left( \begin{matrix} a \\ b \\ 2 \end{matrix} \right) \)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\({ r }=\left( \begin{matrix} 1 \\ -1 \\ -2 \end{matrix} \right) +3 \left( \begin{matrix} -1 \\ -2 \\ 2 \end{matrix} \right) =\left( \begin{matrix} -2 \\ -7 \\ 4 \end{matrix} \right) \)</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>The vector equation of a line L is <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} -1 \\ -2 \\ 3 \end{matrix} \right) +\mu \left( \begin{matrix} 1 \\ -2 \\ -4 \end{matrix} \right) \)</span></p><p>The vector <span class="math-tex">\(\left( \begin{matrix} a \\ b \\ 8 \end{matrix} \right) \)</span> is parallel to this line.</p><p>Find <strong><em>a </em></strong>and <strong><em>b</em></strong></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="4"> <span class="review"></span></p></div><div class="q-explanation"><p>The line <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} { x }_{ 0 } \\ { y }_{ 0 } \\ { z }_{ 0 } \end{matrix} \right) +\mu \left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</span>is parallel to <span class="math-tex">\(\left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</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> 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> <hr class="hidden-separator"> <p>The following quiz tests your understanding of converting between the different forms of a straight line (vector, parametric and Cartesian).</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#bae38dfe"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="bae38dfe"> <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%;"> Equation of Line (different forms) HL <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="11-183-651" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The vector equation of a line is <span class="math-tex">\({ r }=\left( \begin{matrix} -1 \\ 0 \\ 2 \end{matrix} \right) +\mu \left( \begin{matrix} 3 \\ -2 \\ -1 \end{matrix} \right) \)</span>. Which is the correct parametric form of the line.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(x=-1+3\mu \\ \\ z=2-\mu \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(x=-1+2\mu \\ y=3-2\lambda-\mu\\ \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(x=3-\mu \\ y=-2\\ z=-1+2\mu \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(x=-1+3\mu \\ y=-2\mu \\ z=2-\mu \)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} { x }_{ 0 } \\ { y }_{ 0 } \\ { z }_{ 0 } \end{matrix} \right) +\mu \left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</span>corresponds to <span class="math-tex">\(\frac { x-{ x }_{ 0 } }{ l } =\frac { y-{ y }_{ 0 } }{ m } =\frac { z-{ z }_{ 0 } }{ n } \)</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>The vector equation of a line L is <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ -3 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} 2 \\ -3 \\ 1 \end{matrix} \right) \)</span></p><p>The Cartesian equation of a line L is <span class="math-tex">\(\frac { x-a }{ 2 } =\frac { y-b }{ -3 } =\frac { z-c }{ 1 } \)</span></p><p>Find <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</em></strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} { x }_{ 0 } \\ { y }_{ 0 } \\ { z }_{ 0 } \end{matrix} \right) +\mu \left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</span> corresponds to <span class="math-tex">\(\frac { x-{ x }_{ 0 } }{ l } =\frac { y-{ y }_{ 0 } }{ m } =\frac { z-{ z }_{ 0 } }{ n } \)</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>The vector equation of a line L is <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ -1 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} a \\ b \\ c \end{matrix} \right) \)</span></p><p>The Cartesian equation of a line L is <span class="math-tex">\(\frac { 1-x }{ 3 } =\frac { y+1 }{ 2 } =z\)</span></p><p>Find <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</em></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="2"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} { x }_{ 0 } \\ { y }_{ 0 } \\ { z }_{ 0 } \end{matrix} \right) +\mu \left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</span> corresponds to <span class="math-tex">\(\frac { x-{ x }_{ 0 } }{ l } =\frac { y-{ y }_{ 0 } }{ m } =\frac { z-{ z }_{ 0 } }{ n } \)</span></p><p>The Cartesian form can be written as follows <span class="math-tex">\(\frac { x-1 }{ -3 } =\frac { y-(-1) }{ 2 }=\frac { z-0 }{ 1 }\)</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>The Cartesian equation of a line L is <span class="math-tex">\(\frac { x-3 }{ 5 } =2y =z+1\)</span></p><p>The vector equation of a line L is <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} a \\ b \\ c \end{matrix} \right) +\mu \left( \begin{matrix} d \\ e \\ f \end{matrix} \right) \)</span></p><p>Find <strong><em>a </em></strong>, <strong><em>b</em></strong> , <strong><em>c</em></strong> , <strong><em>d</em></strong> , <strong><em>e</em></strong> and <strong><em>f</em></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="0"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>e = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p><p>f = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} { x }_{ 0 } \\ { y }_{ 0 } \\ { z }_{ 0 } \end{matrix} \right) +\mu \left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</span>corresponds to <span class="math-tex">\(\frac { x-{ x }_{ 0 } }{ l } =\frac { y-{ y }_{ 0 } }{ m } =\frac { z-{ z }_{ 0 } }{ n } \)</span></p><p>The Cartesian form can be written as follows <span class="math-tex">\(\frac { x-3 }{ 5 } =\frac { y-0 }{ 0.5 }=\frac { z-(-1) }{ 1 }\)</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>The Cartesian equation of a line L is <span class="math-tex">\(3x=\frac { y+1 }{ 2 } =1+z\)</span></p><p>The line is parallel to the vector <span class="math-tex">\(\left( \begin{matrix} a \\ b \\ -3 \end{matrix} \right) \)</span>.</p><p>Find <strong><em>a </em></strong>and <strong><em>b</em></strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac { x-{ x }_{ 0 } }{ l } =\frac { y-{ y }_{ 0 } }{ m } =\frac { z-{ z }_{ 0 } }{ n } \)</span> corresponds to <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} { x }_{ 0 } \\ { y }_{ 0 } \\ { z }_{ 0 } \end{matrix} \right) +\mu \left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</span>which is parallel to <span class="math-tex">\(\left( \begin{matrix} l \\ m \\ n \end{matrix} \right) \)</span></p><p>The Cartesian form can be written as follows <span class="math-tex">\(\frac { x-0 }{ \frac { 1 }{ 3 } } =\frac { y-(-1) }{ 2 }=\frac { z-(-1) }{ 1 }\)</span> which is parallel to <span class="math-tex">\(\left( \begin{matrix} \frac{1}{3} \\ 2 \\ 1 \end{matrix} \right) \)</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> </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="326"> <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 L passes through the points A(1,-1,3) and B(3,-4,4)</p> <p>Point C (x,y,1) also lies on the line L. Find x and y.</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">If the equation of the straight line is given by <span class="math-tex">\(\textbf{r}= \textbf{a}+\lambda \textbf{b}\)</span> then a certain value of <span class="math-tex">\(\lambda \)</span> will define the position of the position vector <span class="math-tex">\(\overrightarrow { OC } \)</span>. Find this value and use it to find x and y.</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/lines/esq_eqofline2hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/esq_eqofline2hl.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="328"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL 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 line L passes through the points A(0,2,-4) and B(3,-3,2)</p> <p>Point C also lies on the line L. Find the coordinates of C given that <span class="math-tex">\(\left| \overrightarrow { AC } \right| =\left| \overrightarrow { AB } \right| \)</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>Draw a diagram!</p> <p>Find the equation of the straight line. What is the value of <span class="math-tex">\(\lambda \)</span> that defines the position of B? Think about what this value should be for C.</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/lines/esq_eqofline3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/esq_eqofline3.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 3</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="329"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL 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 line L passes through the points A(0,2,-4) and B(3,-3,2)</p> <p>Point C also lies on the line L. Find the possible coordinates of C given that <span class="math-tex">\(\left| \overrightarrow { AC } \right| =2\left| \overrightarrow { AB } \right| \)</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>Draw a diagram!</p> <p>Is there just one answer?</p> <p>Find the equation of the straight line. What is the value of <span class="math-tex">\(\lambda \)</span> that defines the position of B? Think about what this value should be for C.</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/lines/esq_eqofline4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/esq_eqofline4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="651"> <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>Equation of a Line</strong> have you 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