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onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Equations of Planes</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/planes/main.jpg" style="float: left; width: 100px; height: 100px;">This is a hugely important topic in the HL course. We can describe planes in 3D in a number of different ways. It is not enough to learn the different equations, but it is vital that you have a strong conceptual understanding of the different forms of the equations of the plane, in particular, the <strong><em>normal form</em></strong>. You will need to be familiar with the Vector Product and the Scalar Product before you start this topic. Questions in the exam on this topic are often long questions from section B.</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 planes in 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>Planes - Vector, Normal & Cartesian Forms</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="312"> <p>In the following video we are going to look at the equations of planes, in particular the three different forms: Vector, Normal and Cartesian.</p> <table align="center" border="0" cellpadding="0" cellspacing="0" style="width: 70%;"> <tbody> <tr> <td style="text-align: center;"><strong>r</strong> =<strong> a</strong> + λ<strong>b</strong> + μ<strong>c</strong></td> <td style="text-align: center;">Vector Form</td> </tr> <tr> <td style="text-align: center;"><strong>r∙n </strong>= <strong>a∙n</strong></td> <td style="text-align: center;">Normal Form</td> </tr> <tr> <td style="text-align: center;">ax + by + cz = d</td> <td style="text-align: center;">Cartesian Form</td> </tr> </tbody> </table> <p>We will find out how to convert from one form to another by looking at the follwing example:</p> <p><em>Convert the following into normal and Cartesian form</em></p> <p style="text-align: center;"><em><span class="math-tex">\(\textbf{r} =\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\lambda \left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right) +\mu \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</span></em></p> <p style="text-align: center"><iframe allowfullscreen="" frameborder="0" height="360" mozallowfullscreen="" src="https://player.vimeo.com/video/250778867" webkitallowfullscreen="" width="640"><br> </iframe></p> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/equation-of-planes---vector_-normal-and-cartesian-form.pdf">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/equation-of-planes---vector_-normal-and-cartesian-form.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"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Equation of a Plane from 3 Points</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="313"> <p>In the following video we are going to try to find the equation of a plane that is formed by three points. Also, we will be able to see if a 4th point lies in the same plane (Are the four points coplanar?)</p> <p><em>Find the equation of the plane formed by the triangle A(1 , 2 , -1) , B(2 , -2 , 3) and C(0 , 2 , 1)</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/265027553"></iframe></div> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Notes from video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/a-plane-from-3-points.pdf" onclick="window.open(this.href, '', 'resizable=no,status=no,location=no,toolbar=no,menubar=no,fullscreen=no,scrollbars=no,dependent=no'); return false;">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/a-plane-from-3-points.pdf" width="640"></iframe></p> </section> <p><em>Does a 4th point D (1 , -1 , 2) lie in the plane?</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/265136269"></iframe></div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/vectors/planes/planes_revision-notes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/planes/planes_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 short quiz to practise the skills you have learned in this topic.</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#1aa87177"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="1aa87177"> <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 Planes <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="11-179-650" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is the equation of a plane?</p><p><em>Find <strong>all</strong> correct answers</em></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 2 \\ -1 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} -1 \\ 1 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\frac { x-2 }{ 3 } =\frac { y+5 }{ -2 } =z\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 2 \\ -1 \\ 0 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</span></label></p><p><label class="checkbox"><input type="checkbox"> x + 2y = 9</label></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac { x-2 }{ 3 } =\frac { y+5 }{ -2 } =z\)</span> and <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 2 \\ -1 \\ 0 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</span> are equations of planes in 3D</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is a point that lies on the plane <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ -1 \\ 3 \end{matrix} \right) +\mu \left( \begin{matrix} -2 \\ 0 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} -2 \\ 0 \\ 3 \end{matrix} \right) \)</span> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> (-2,0,1)</label></p><p><label class="radio"><input type="radio"> (1,-2,-2)</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left( \begin{matrix} -2 \\ 0 \\ 1 \end{matrix} \right) \times \left( \begin{matrix} -2 \\ 0 \\ 3 \end{matrix} \right) \)</span></label></p><p><label class="radio"><input class="c" type="radio"> (1,-1, 3)</label></p></div><div class="q-explanation"><p>The vector form of a plane is given in the form <strong>r</strong>=<strong>a</strong>+λ<strong>b</strong>+μ<strong>c</strong> where <strong>a </strong>is a point that lies in the plane.</p><p>Therefore the position vector <span class="math-tex">\(\left( \begin{matrix} 1 \\ -1 \\ 3 \end{matrix} \right)\)</span> is a point on the plane</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is a point that lies on the plane</p><p>x = 1 + 2λ - 2μ</p><p>y = -1 - 1λb + 4μ</p><p>z = 4 + 3λb</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> (1,2,-2)</label></p><p><label class="radio"><input type="radio"> This is not an equation of a plane</label></p><p><label class="radio"><input class="c" type="radio"> (1,-1,4)</label></p><p><label class="radio"><input type="radio"> (2, -1, 3)</label></p></div><div class="q-explanation"><p>This is the parametric form of the equation of a plane. Try writing it in vector form.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\left( \begin{matrix} a \\ b \\ c \end{matrix} \right) \)</span>is a vector perpendicular to the plane x + 2y - 3z = 8. Find a, b and c.</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><p>c = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> </p></div><div class="q-explanation"><p>The plane here is given in Cartesian form. The coefficients of x, y and z give us the normal to the vector. Any multiple of the vector <span class="math-tex">\(\left( \begin{matrix} 1 \\ 2 \\ -3 \end{matrix} \right) \)</span>is perpendicular to the plane.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following vectors is a normal to the plane</p><p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ -1 \\ 3 \end{matrix} \right) +\mu \left( \begin{matrix} -2 \\ 0 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} -2 \\ 0 \\ 3 \end{matrix} \right) \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\left( \begin{matrix} -2 \\ 0 \\ 1 \end{matrix} \right) \times \left( \begin{matrix} -2 \\ 0 \\ 3 \end{matrix} \right) \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left( \begin{matrix} 1 \\ -1 \\ 3 \end{matrix} \right) \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left( \begin{matrix} 1 \\ -2 \\ -2 \end{matrix} \right) \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left( \begin{matrix} -2 \\ 0 \\ 1 \end{matrix} \right) \)</span></label></p></div><div class="q-explanation"><p>To find a vector perpendicular to a plane given in vector form, we find the vector product of the two direction vectors.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\left( \begin{matrix} 8 \\ a \\ b \end{matrix} \right) \)</span>is a normal to the plane <span class="math-tex">\({ r }=\left( \begin{matrix} 0 \\ 1 \\ -2 \end{matrix} \right) +\mu \left( \begin{matrix} 2 \\ -1 \\ 4 \end{matrix} \right) +\lambda \left( \begin{matrix} -1 \\ 2 \\ 0 \end{matrix} \right) \)</span></p><p>Find a and b</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p>Find the vector product of the two direction vectors. <span class="math-tex">\(\left( \begin{matrix} 2 \\ -1 \\ 4 \end{matrix} \right) \times \left( \begin{matrix} -1 \\ 2 \\ 0 \end{matrix} \right) =\left( \begin{matrix} -8 \\ -4 \\ 3 \end{matrix} \right) \)</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"><div class="q-question"><p>Which of the following planes are parallel</p><p><span class="math-tex">\({ \Pi }_{ 1 }:2x-y+z=8\)</span></p><p><span class="math-tex">\({ \Pi }_{ 2 }:x+2y=8\)</span></p><p><span class="math-tex">\({ \Pi }_{ 3 }:-4x+2y+2z=3\)</span></p></div></div><div class="q-answer"><p><label class="radio"><input type="radio"> None are parallel</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ \Pi }_{ 2 } \quad and \quad { \Pi }_{ 3 }\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\({ \Pi }_{ 1 } \quad and \quad { \Pi }_{ 3 }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ \Pi }_{ 1 } \quad and \quad { \Pi }_{ 2 }\)</span></label></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following planes are perpendicular</p><p><span class="math-tex">\({ \Pi }_{ 1 }:2x-y+z=8\)</span></p><p><span class="math-tex">\({ \Pi }_{ 2 }:x+2y=8\)</span></p><p><span class="math-tex">\({ \Pi }_{ 3 }:2x-y-3z=3\)</span></p><p><em>Find <strong>all</strong> correct answers</em></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\({ \Pi }_{ 1 } \quad and \quad { \Pi }_{ 2 }\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\({ \Pi }_{ 1 } \quad and \quad { \Pi }_{ 3 }\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\({ \Pi }_{ 2 } \quad and \quad { \Pi }_{ 3 }\)</span></label></p><p><label class="checkbox"><input type="checkbox"> None of the planes are perpendicular</label></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 2 \\ -1 \\ 1 \end{matrix} \right) \cdot \left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) =0\quad \quad \left( \begin{matrix} 2 \\ -1 \\ 1 \end{matrix} \right) \cdot \left( \begin{matrix} 2 \\ -1 \\ -3 \end{matrix} \right) =3\quad \quad \left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) \cdot \left( \begin{matrix} 2 \\ -1 \\ -3 \end{matrix} \right) =0\\ \)</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="317"> <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 plane has vector equation <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</span></p> <p>Show that the Cartesian equation of the plane is x - 5y + 3z + 9 = 0</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>Find the normal to the plane by finding the vector product of <span class="math-tex">\(\left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right)\)</span> and <span class="math-tex">\(\left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/eqplane4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane4.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="314"> <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 coordinates of points A, B and C are given as (5,4,1) , (5,1,-2) and (1,-1,2) respectively.</p> <p>a) Find the equation of the plane that passes through A, B and C</p> <p>b) Find the equation of the plane that is perpendicular to AB and passes through C</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) The normal to the plane can be found from <span class="math-tex">\(\overrightarrow { AB } \times \overrightarrow { AC } \)</span></p> <p>b) The normal to the plane is vector <span class="math-tex">\(\overrightarrow { AB }\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/eqplane1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-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="315"> <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>A plane has vector equation <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} -2 \\ 0 \\ 5 \end{matrix} \right) +\lambda \left( \begin{matrix} 0 \\ -4 \\ 5 \end{matrix} \right) \)</span></p> <p>a) Find the Cartesian equation of the plane</p> <p>b) The plane meets the x, y and z axes at A, B and C respectively. OABC forms a pyramid. Find the volume of the pyramid.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) The normal to the plane can be found from the vector product of the two directions and the point (1 , 2 , 0).</p> <p>b) The plane crosses the x axis when y = z = 0. Find the corrdinates of A. Similarly, find coordinates of B and 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/planes/eqplane2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane2.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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="316"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the Cartesian equation of the plane that is perpendicular to the plane 2x - y + z = 8 and contains the points A(4,2,-3) and B(6,1,-1).</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>The plane is perpendicular to the normal and the vector <span class="math-tex">\(\overrightarrow { AB } \)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/eqplane3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="650"> <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>Equations of Planes</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your 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