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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 Line and Plane</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/line-and-plane/main.jpg" style="float: left; width: 100px; height: 99px;"></p> <p>For questions involving lines and planes, we are usually asked to find the point of intersection. However, when we consider a line and a plane, there are three possible situations. In this page, we will consider all the possible outcomes.</p> <p style="text-align: center;">1) the line intersects the plane</p> <p style="text-align: center;"> 2) the line is parallel to the plane</p> <p style="text-align: center;"> 3) the line lies in the plane</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>the intersection of a line and a plane</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>A Line and a Plane</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="457"> <p>In the following video, we consider the three different possibilities</p> <h4>Example 1 - Point of Intersection</h4> <p>Find the point of intersection of the line <span class="math-tex">\(-x=\frac {y-5}{2}=2z-8\)</span> with the plane 3x – y + z = 8</p> <h4>Example 2 - No Intersection</h4> <p>Find the point of intersection of the line <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2 \\ 1\\0 \end{matrix} \right) +\mu\left( \begin{matrix} 4 \\ 3 \\1\end{matrix} \right) \)</span>with the plane x - 2y + 2z = 6</p> <h4>Example 3 - Line lies in Plane</h4> <p>Find the point of intersection of the line <span class="math-tex">\(x=2+4λ\ ,\ y=1+3λ\ ,\ z=λ\)</span> with the plane x - 2y + 2z = 0</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/249634696"></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/line-and-plane/intersection-of-a-line-and-a-plane.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/intersection-of-a-line-and-a-plane.pdf" width="640"></iframe></p> </section> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <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>Reflecting a Point in a Plane</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="462"> <p>In the following video, we look at a specific application of finding the intersection of a line with a plane. We will see how it is possible to find the coordinates of the point reflected in a plane. In order to do this, we have to find the equation of a line that goes through our point and is perpendicular to the plane.</p> <p style="text-align: center;"><img alt="" src="../../files/vectors/line-and-plane/esq4.jpg" style="width: 400px; height: 401px;"></p> <p>Here is the example:</p> <hr> <p><em><strong>The point A (1, 3, 0) is on the line L, which is perpendicular to the plane <span class="math-tex">\(3x−3y+2z=5\)</span>.</strong></em></p> <ol style="list-style-type:lower-alpha;"> <li><em><strong>Find the equation of the line L.</strong></em></li> <li><em><strong>Find the point R which is the intersection of the line L and the plane.</strong></em></li> <li><em><strong>The point A is reflected in the plane. Find the coordinates of the image of A.</strong></em></li> </ol> <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/281218314"></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/line-and-plane/relecting-a-point-in-a-plane.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/relecting-a-point-in-a-plane.pdf" width="640"></iframe></p> </section> </div> <p> </p> </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/line-and-plane/revision-notes_intersection_of_line_and_plane.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/line-and-plane/revision-notes_intersection_of_line_and_plane.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="#c55696fa"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="c55696fa"> <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 line and plane <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="11-204-666" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Find the value of <span class="math-tex">\(\lambda\)</span> in order that the point (6,1,-1) lies on the line <span></span><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2\\3 \\ -1 \end{matrix} \right) +\lambda \left( \begin{matrix} -2 \\ 1\\0 \end{matrix} \right) \)</span><span></span></p></div><div class="q-answer"><p><span class="math-tex">\(\lambda\)</span> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 2\\3 \\ -1 \end{matrix} \right) -2 \left( \begin{matrix} -2 \\ 1\\0 \end{matrix} \right) =\left( \begin{matrix} 6 \\ 1\\-1 \end{matrix} \right)\)</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>The point (2 , <strong><em>a</em></strong> , <strong><em>b </em></strong>) lies on the line <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} -1\\0 \\ 2 \end{matrix} \right) +\lambda \left( \begin{matrix} 1 \\ -1\\2 \end{matrix} \right) \)</span></p><p>Find <strong><em>a </em></strong>and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a </em></strong>= <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p><strong><em>b </em></strong>= <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} -1\\0 \\ 2 \end{matrix} \right) +3 \left( \begin{matrix} 1 \\ -1\\2 \end{matrix} \right) =\left( \begin{matrix} 2 \\ a\\b \end{matrix} \right) \)</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 point (1 , 2 , -1) lies in the plane x + 2y - 3z = <strong><em>a</em></strong></p><p>Find <strong><em>a</em></strong>.</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>x + 2y - 3z = <strong><em>a</em></strong></p><p>(1) + 2(2) - 3(-1) = <strong><em>a</em></strong></p><p>8 = <strong><em>a</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>The point (3 , 0 , <strong><em>k</em></strong>) lies in the plane 2x - y + 3z - 3 = 0</p><p>Find <strong><em>k</em></strong>.</p></div><div class="q-answer"><p><em><strong>k</strong></em> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>2x - y + 3z -3 = 0</p><p>2(3) - (0) + 3<strong><em>k</em></strong> - 3 = 0</p><p>3<strong><em>k</em></strong> = -3</p><p><strong><em>k</em></strong> = -1</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 line <strong><em>L</em></strong> does <strong>not</strong> intersect with the plane 4x + y + <strong><em>k</em></strong>z = 3 . Find <strong><em>k</em></strong></p><p><strong><em>L</em></strong>: <span class="math-tex">\(x = 1+2\lambda\\
y = 3+\lambda\\
z=4-\lambda \)</span></p></div><div class="q-answer"><p><strong><em>k</em></strong>= <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p>The line and plane must be parallel.</p><p>This means that the line is perpendicular to the normal.</p><p>The scalar product of the vectors equals zero.</p><p><span></span><span class="math-tex">\(\left( \begin{matrix} 2 \\ 1\\-1 \end{matrix} \right)\cdot \left( \begin{matrix} 4 \\ 1\\k \end{matrix} \right)=0\)</span><span></span></p><p>8 + 1 - <em><strong>k</strong></em> = 0</p><p><strong><em>k</em></strong> = 9</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>(0 , <strong><em>a </em></strong>, <strong><em>b</em></strong>) is the point where the line line <span></span><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2\\1 \\ 4 \end{matrix} \right) +\lambda\left( \begin{matrix} -1 \\ 2\\3 \end{matrix} \right) \)</span><span> meets the plane x = 0.</span></p><p>Find <strong><em>a </em></strong> and <strong><em>b</em></strong>.</p></div><div class="q-answer"><p><strong><em>a </em></strong>= <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p><strong><em>b </em></strong>= <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 2\\1 \\ 4 \end{matrix} \right) +\lambda\left( \begin{matrix} -1 \\ 2\\3 \end{matrix} \right) =\left( \begin{matrix} 0 \\ a\\b \end{matrix} \right) \)</span></p><p><span class="math-tex">\(\left( \begin{matrix} 2\\1 \\ 4 \end{matrix} \right) +2\left( \begin{matrix} -1 \\ 2\\3 \end{matrix} \right) =\left( \begin{matrix} 0 \\ a\\b \end{matrix} \right) \)</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 line <span class="math-tex">\(x=1+\lambda\\
y=1-\lambda\\
z=2+\lambda\)</span> intersects with the plane x + y + z = 2 at the point (<em><strong>a</strong></em> , <em><strong>b</strong></em> , <em><strong>c</strong></em>).</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><strong><em>a </em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong><em>c</em></strong> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(x + y + z = 2 \\
(1+\lambda)+(1-\lambda)+(2+\lambda)=2\\
4+\lambda = 2\\
\lambda = -2\)</span></p><p>x = 1 + (-2)</p><p>y = 1 - (-2)</p><p>z = 2 + (-2)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The plane <strong><em>a</em></strong>x + 2y + 1z = 3 is perpendicular to the line <span class="math-tex">\(\frac{x-1}{2}=y+1=2z\)</span>.</p><p>Find <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>If the plane and line are perpendicular, then the normal is parallel to the line.</p><p>The direction of the line is <span class="math-tex">\(\left( \begin{matrix} 2 \\ 1\\0.5 \end{matrix} \right)\)</span></p><p>The direction of the normal is <span class="math-tex">\(\left( \begin{matrix} a \\ 2\\1 \end{matrix} \right)\)</span></p><p><span></span><span class="math-tex">\(\left( \begin{matrix} a \\ 2\\1 \end{matrix} \right) =k\left( \begin{matrix} 2 \\ 1\\0.5 \end{matrix} \right)\)</span><span></span></p><p><span class="math-tex">\(\left( \begin{matrix} a \\ 2\\1 \end{matrix} \right) =2\left( \begin{matrix} 2 \\ 1\\0.5 \end{matrix} \right)\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the intersection of the line <strong><em>L</em></strong> and the plane 3x - 2y + z = - 12 by filling in the blanks below.</p><p><span class="math-tex">\(L:\textbf{r}=\left( \begin{matrix} 1\\0 \\ -5 \end{matrix} \right) +\lambda\left( \begin{matrix} -2 \\ 1\\3 \end{matrix} \right) \)</span></p></div><div class="q-answer"><p><span class="math-tex">\(x = 1 - 2\lambda\\
y = \lambda\\
z = -5+3\lambda\\\)</span></p><hr><p><span class="math-tex">\(3(1 - 2\lambda)-2\lambda+(-5+3\lambda)=-12\)</span></p><p><span class="math-tex">\(3 - 6\lambda-2\lambda-5+3\lambda=-12\)</span></p><p><span class="math-tex">\(-5\lambda=\)</span><input type="text" style="height: auto;" data-c="-10"> <span class="review"></span></p><p><span class="math-tex">\(\lambda = \)</span> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><hr><p>x = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p>y = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>z = <input type="text" style="height: auto;" data-c="1"> <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 <strong><em>L</em></strong> is perpendicular to the plane <span class="math-tex">\(\Pi\)</span></p><p>The point of intersection of the line and the plane is (-2 , 1 , 2).</p><p>The point A(1 , 2 , 0) lies on the line <strong><em>L</em></strong>.</p><p>The reflection of the point A in the plane <span class="math-tex">\(\Pi\)</span> is (<em><strong>a</strong></em> , <em><strong>b</strong></em> ,<em><strong> c</strong></em>)</p><p>Find <em><strong>a</strong></em> , <em><strong>b</strong></em> and<em><strong> c</strong></em></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p><p><strong><em>c</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/vectors/line-and-plane/quiz10.jpg" style="width: 500px; height: 469px;"></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="458"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The points A and B are given by A(-8,1,-2) and B(-2,-1,2).</p> <p>A plane Π is defined by the equation <span class="math-tex">\(2x−y−3z=−8\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find a vector equation of the line L passing through the points A and B.</li> <li>Find the coordinates of the point of intersection of the line and the plane.</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>To find the equation of the line, you will need to find <span class="math-tex">\(\overrightarrow{AB}\)</span></p> <p><span class="math-tex">\(\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}\)</span><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/line-and-plane/esq_line_and_plane1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane1.pdf" width="640"></iframe></p> </section> </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="459"> <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>Consider the plane <span class="math-tex">\(x−2y+4z=−15\)</span> and the line</p> <p><span class="math-tex">\(x=3+kλ\\ y=−2+λ\\ z=(2k+6)−2λ\)</span></p> <p>The line and the plane are perpendicular. Find</p> <ol style="list-style-type:lower-alpha;"> <li>The value of <strong><em>k</em></strong></li> <li>The coordinates of the point of intersection of the line and the plane.</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 lang="fr" style="margin:0in;font-family:Calibri;font-size:14.0pt">If line and plane are <span style="font-weight:bold">perpendicular</span>, then line is <span style="font-weight:bold">parallel</span> to normal to the plane</p> <p><content><img alt="" src="../../files/vectors/line-and-plane/esq2.jpg" style="width: 300px; height: 199px;"> </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/line-and-plane/esq_line_and_plane2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-default 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="460"> <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><span class="math-tex">\(Π_{ 1 }\)</span>and <span class="math-tex">\(Π_{ 2 }\)</span> are planes such that</p> <p><span class="math-tex">\(Π_{ 1 }:2x−y−2z=0\)</span></p> <p>and</p> <p><span class="math-tex">\(Π_{ 2 }:−2x+3y+3z=4 \)</span></p> <p><em><strong>L</strong></em> is the intersection of planes <span class="math-tex">\(Π_{ 1 }\)</span> and <span class="math-tex">\(Π_{ 2 }\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the equation of the line</li> </ol> <p>A third plane <span class="math-tex">\(Π_{ 3 }\)</span> is defined by the equation <span class="math-tex">\(kx+(k−1)y−z=5\)</span></p> <ol style="list-style-type:lower-alpha;"> <li value="2">Find the value of <strong><em>k </em></strong>such that the line <strong><em>L</em></strong> does <strong>not</strong> intersect with <span class="math-tex">\(Π_{ 3 }\)</span></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>If the line <strong><em>L</em></strong> does not intersect with <span class="math-tex">\(Π_{ 3 }\)</span> then they must be parallel</p> <p><img alt="" src="../../files/vectors/line-and-plane/esq3.jpg" style="width: 252px; height: 247px;"></p> <p>Here is a graph of the three planes and the line</p> <p><content> </content><img alt="" src="../../files/vectors/line-and-plane/esq3a.jpg" style="width: 500px; height: 390px;"></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/line-and-plane/esq_line_and_plane3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane3.pdf" width="640"></iframe></p> </section> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="461"> <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 point A (3, 1, –2) is on the line L, which is perpendicular to the plane <span class="math-tex">\(2x−3y−z+9=0\)</span>.</p> <ol style="list-style-type:lower-alpha;"> <li>Find the Cartesian equation of the line L.</li> <li>Find the point R which is the intersection of the line L and the plane.</li> <li>The point A is reflected in the plane. Find the coordinates of the image of A.</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 points A , R and A' lie on the straight line:</p> <p><content> </content><img alt="" src="../../files/vectors/line-and-plane/esq4.jpg" style="width: 400px; height: 401px;"></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/line-and-plane/esq_line_and_plane4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane4.pdf" 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