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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">Intersection of Lines</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/intersection_lines/main.jpg" style="float: left; width: 100px; height: 101px;"> This page looks at the intersections of lines in 2 and 3 dimensions. In 2D, lines intersect or are parallel. In 3D, it is slightly more complicated, they can intersect or be parallel or be skew (not parallel and not intersect!). We will also look at applications of this with moving objects (<a href="../653/kinematics.html" title="Kinematics">Kinematics</a>) in which boats and planes can cross paths and collide.</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>coincident, parallel, intersecting and skew lines</li> <li>the point of intersection of two lines</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>Intersection of 2 Lines</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="447"> <p>In the following video, we are going to look at the intersection of two lines. We will consider the following example</p> <p>Find the coordinates of the point of intersection of the following two lines</p> <p><span class="math-tex">\(\textbf{r}_1=\left( \begin{matrix} 1 \\ 0 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} 2 \\ -3 \\ 1 \end{matrix} \right) \)</span></p> <p><span class="math-tex">\(\textbf{r}_2=\left( \begin{matrix} 5 \\ 1 \\ 0 \end{matrix} \right) +\mu\left( \begin{matrix} 1 \\ 2 \\ -1 \end{matrix} \right) \)</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/249774418"></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/intersection_lines/intersection-of-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/intersection_lines/intersection-of-2-lines.pdf" width="640"></iframe></p> </section> </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>Kinematics and Collisions</p> </div> </div> <div class="panel-body"> <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 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/intersection_lines/intersecting_lines_revision-notes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/intersection_lines/intersecting_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 that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#e10b8c56"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="e10b8c56"> <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%;"> Intersection of Lines <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="11-203-667" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><span>Below are the equations of two different lines:</span><span></span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(A: \textbf{r}=\left( \begin{matrix} 1 \\ -1 \end{matrix} \right) +\lambda\left( \begin{matrix} -2 \\ 1 \end{matrix} \right) \)</span></td><td style="text-align: center;"><span class="math-tex">\(B: \textbf{r}=\left( \begin{matrix} 2 \\ 3 \end{matrix} \right) +\lambda\left( \begin{matrix} 1 \\ 3 \end{matrix} \right) \)</span></td></tr></tbody></table></div><div class="q-answer"><p>Which of the above lines intersects with <span class="math-tex">\( \textbf{r}=(-1+2\mu) \textbf{i}\ +(1-6\mu) \textbf{j}\)</span> <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p><p>Which of the above lines intersects with <span class="math-tex">\( \textbf{r}=(-1-4\mu) \textbf{i}\ +(2\mu) \textbf{j}\)</span> <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>Which of the above lines is parallel to <span class="math-tex">\( \textbf{r}=(2+6\mu) \textbf{i}\ +(-1-3\mu) \textbf{j}\)</span> <input type="text" style="height: auto;" data-c="A"> <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>The line <span class="math-tex">\( \textbf{r}=(-2+3\lambda) \textbf{i}\ +(-6-2\lambda) \textbf{j}+ (4\lambda)\textbf{k}\)</span> intersects with only one of the lines below. Which one?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 1 \\ -1\\8 \end{matrix} \right) +\mu\left( \begin{matrix} 1 \\ -3 \\0\end{matrix} \right) \)</span></label><label class="radio"> </label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 1 \\ 3\\0 \end{matrix} \right) +\mu\left( \begin{matrix} -3 \\ 2 \\-4\end{matrix} \right) \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 0 \\ 0\\0 \end{matrix} \right) +\mu\left( \begin{matrix} 6 \\ -2 \\8\end{matrix} \right) \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 1 \\ -3\\0 \end{matrix} \right) +\mu\left( \begin{matrix} -9 \\ 6 \\-12\end{matrix} \right) \)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 1 \\ -1\\8 \end{matrix} \right) +\mu\left( \begin{matrix} 1 \\ -3 \\0\end{matrix} \right) \)</span> is the only line that is not parallel to <span class="math-tex">\( \textbf{r}=(-2+3\lambda) \textbf{i}\ +(-6-2\lambda) \textbf{j}+ (4\lambda)\textbf{k}\)</span></p><p>The lines intersect at (4,-10,8)</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 following two boats collide at t=5.</p><p><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} a \\ 0\\-1 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -1 \\1\end{matrix} \right) \)</span></p><p><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 3 \\ b\\9 \end{matrix} \right) +t\left( \begin{matrix} 1 \\ 1 \\c\end{matrix} \right) \)</span></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="-10"> <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">\(\left( \begin{matrix} -2 \\ 0\\-1 \end{matrix} \right) +5\left( \begin{matrix} 2 \\ -1 \\1\end{matrix} \right)=\left( \begin{matrix} 8 \\ -5\\4 \end{matrix} \right) =\left( \begin{matrix} 3 \\ -10\\9 \end{matrix} \right) +5\left( \begin{matrix} 1 \\ 1 \\-1\end{matrix} \right) \)</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>Find the time taken for <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 5 \\- 10\\2 \end{matrix} \right) +t\left( \begin{matrix} -1 \\ 2 \\0\end{matrix} \right) \)</span> to reach (-1,2,2)</p></div><div class="q-answer"><p> t = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 5 \\- 10\\2 \end{matrix} \right) +6\left( \begin{matrix} -1 \\ 2 \\0\end{matrix} \right) =\left( \begin{matrix} -1 \\ 2 \\2\end{matrix} \right) \)</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>Find <strong><em>a </em></strong>if the two paths do not cross</p><p><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} 5\\-1 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)</span> , <span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 0\\-2 \end{matrix} \right) +t\left( \begin{matrix} 4 \\ a \end{matrix} \right) \)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p>Paths need to be parallel <span class="math-tex">\(\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) =k\left( \begin{matrix} 4 \\a \end{matrix} \right) \\ k=-2\)</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>Two objects have position vectors given by <span class="math-tex">\(\textbf{r}_{A}\)</span> and <span class="math-tex">\(\textbf{r}_{B}\)</span> , where <strong><em>t</em></strong> is time</p><p><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} -4\\-1 \end{matrix} \right) +t\left( \begin{matrix} 3 \\ -1 \end{matrix} \right) \)</span></p><p><span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 17\\-9 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 1 \end{matrix} \right) \)</span></p><p>Show that the objects do not collide by completeing the solution below</p></div><div class="q-answer"><p>Drag the answers into the correct places</p><p> <span class="q-text-draggable draggable" draggable="true">2t</span> <span class="q-text-draggable draggable" draggable="true">-t</span> <span class="q-text-draggable draggable" draggable="true">4.2</span> <span class="q-text-draggable draggable" draggable="true">4</span> <span class="q-text-draggable draggable" draggable="true">t</span> <span class="q-text-draggable draggable" draggable="true">13</span> <span class="q-text-draggable draggable" draggable="true">21</span> <span class="q-text-draggable draggable" draggable="true">8</span> <span class="q-text-draggable draggable" draggable="true">-10</span></p><p>-4 + 3t = 17 - <input type="text" style="height: auto;" data-c=" 2t"> <span class="review"></span> </p><p>5t = <input type="text" style="height: auto;" data-c="21"> <span class="review"></span></p><p>t = <input type="text" style="height: auto;" data-c="4.2"> <span class="review"></span></p><p>-1 - t = -9 + <input type="text" style="height: auto;" data-c=" t"> <span class="review"></span> </p><p>2t = <input type="text" style="height: auto;" data-c=" 8"> <span class="review"></span> </p><p>t = <input type="text" style="height: auto;" data-c=" 4"> <span class="review"></span> </p><p>Since the times are not equal, the objects do not collide.</p></div><div class="q-explanation"><p>-4 + 3t = 17 - 2t</p><p>5t = 21</p><p>t = 4.2</p><p>-1 - t = -9 + t</p><p>8 = 2t</p><p>t = 4</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 class="smart-object center" data-id="453"> <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<sub>1</sub> passes through the points P(-13,-6,1) and Q(3,2,-3).</p> <p>A second line L<sub>2</sub> has equation <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 9\\12 \\ 2 \end{matrix} \right) +s\left( \begin{matrix} -3 \\ 2\\4 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Show that <span class="math-tex">\(\overrightarrow{PQ}=\left( \begin{matrix} 16 \\ 8\\-4 \end{matrix} \right) \)</span></li> <li>Hence, write down the equation L<sub>1</sub> in the form <span class="math-tex">\(\textbf{r}=\textbf{a}+t \textbf{b}\)</span></li> <li>The lines L<sub>1</sub> and L<sub>2</sub> intersect at the point R. Find the coordinates of R.</li> </ol> <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">\(\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}\)</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/intersection_lines/esq_intersections1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections1.pdf" width="640"></iframe></p> </section> </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="454"> <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> <div>The diagram shows quadrilateral ABCD with vertices A(6,0) , B(3,5) , C(-10,4) and D(1,-3).</div> <div><img alt="" src="../../files/vectors/intersection_lines/quadrilateral" style="width: 600px; height: 403px;"></div> <div> <ol style="list-style-type:lower-alpha;"> <li>Find <span class="math-tex">\(\overrightarrow{AC}\)</span></li> <li>Show that <span class="math-tex">\(\overrightarrow{BD}\)</span> is perpendicular to <span class="math-tex">\(\overrightarrow{AC}\)</span></li> <li>Write down the equation of the line (AC) in the form</li> <li>Write down the equation of the line (BD)</li> <li>The lines (AC) and (BD) intersect at E. Find the coordinates of E</li> </ol> </div> <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>b. Use the scalar product to show that the lines are perpendicular</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/intersection_lines/esq_intersections2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-border panel-has-colored-body 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> <div class="smart-object center" data-id="455"> <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>Two boats A and B, move so that a time <em><strong>t</strong></em> hours, their positions, in kilometres, are given by</p> <p><span></span><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)</span><span></span></p> <p><span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the position where the two boats cross.</li> <li>Show that the boats do not collide.</li> </ol> <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>The key to answering this question correctly is to make the assumption that they will have the same position at some time, but that <strong>the time is not the same for the two boats</strong>.</p> <p>Use t<sub>A</sub> and t<sub>B</sub></p> <content> </content></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/intersection_lines/esq_intersections3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> <div class="smart-object center" data-id="455"> <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>Two boats A and B, move so that a time <em><strong>t</strong></em> hours, their positions, in kilometres, are given by</p> <p><span></span><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)</span><span></span></p> <p><span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the position where the two boats cross.</li> <li>Show that the boats do not collide.</li> </ol> <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>The key to answering this question correctly is to make the assumption that they will have the same position at some time, but that <strong>the time is not the same for the two boats</strong>.</p> <p>Use t<sub>A</sub> and t<sub>B</sub></p> <content> </content></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/intersection_lines/esq_intersections3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections3.pdf" width="640"></iframe></p> </section> </div> <div class="smart-object center" data-id="455"> <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>Two boats A and B, move so that a time <em><strong>t</strong></em> hours, their positions, in kilometres, are given by</p> <p><span></span><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)</span><span></span></p> <p><span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the position where the two boats cross.</li> <li>Show that the boats do not collide.</li> </ol> <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>The key to answering this question correctly is to make the assumption that they will have the same position at some time, but that <strong>the time is not the same for the two boats</strong>.</p> <p>Use t<sub>A</sub> and t<sub>B</sub></p> <content> </content></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/intersection_lines/esq_intersections3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections3.pdf" width="640"></iframe></p> </section> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="456"> <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 line L<sub>1</sub> has equation <span></span><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2\\-1 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} -1 \\ -2\\k \end{matrix} \right) \)</span><span></span></p> <p>The line L<sub>2</sub> has equation <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2\\1 \\ -3 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -1\\-1 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>The point A(3,1,-1) lies on the line L<sub>1</sub>. Show that k = 4.</li> <li>Show that the lines and L<sub>1</sub> are L<sub>2</sub> perpendicular.</li> <li>Show that the lines and L<sub>1</sub> do not L<sub>2</sub> intersect.</li> <li>The point B lies on the line The point C has coordinates (2,1,-3). ABC forms an isosceles triangles with AC=BC. Find the coordinates of B.</li> </ol> <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>d. You should be able to work out the length of AC</p> <p>Consider that B lies on the line L<sub>1</sub> . Find the vector <span class="math-tex">\(\overrightarrow{BC}\)</span></p> <p><img alt="" src="../../files/vectors/intersection_lines/esq4.jpg" style="width: 435px; height: 258px;"><content></content></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/intersection_lines/esq_intersections4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections4.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="page-container panel-self-assessment" data-id="667"> <div 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