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</div><h2>SL Paper 2</h2><div class="specification">
<p class="p1">Let \(u = 6i + 3j + 6k\) and \(v = 2i + 2j + k\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p1">(i) <span class="Apple-converted-space"> \(u \bullet v\)</span>;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(\left| {{u}} \right|\);</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>\(\left| {{v}} \right|\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the angle between \({{u}}\) and \({{v}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Two lines with equations \({{\boldsymbol{r}}_1} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>3\\<br>{ - 1}<br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br>5\\<br>{ - 3}\\<br>2<br>\end{array}} \right)\) and \({{\boldsymbol{r}}_2} = \left( {\begin{array}{*{20}{c}}<br>9\\<br>2\\<br>2<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>{ - 3}\\<br>5\\<br>{ - 1}<br>\end{array}} \right)\) intersect at the point P. Find the coordinates of P.</span></p>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 3}\\<br>6<br>\end{array}} \right)\) and \({\boldsymbol{w}} = \left( {\begin{array}{*{20}{c}}<br>k\\<br>{ - 2}\\<br>4<br>\end{array}} \right)\) , for \(k > 0\) . The angle between <strong><em>v</em></strong> and <strong><em>w</em></strong> is \(\frac{\pi }{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(k\) .<br></span></p>
</div>
<br><hr><br><div class="specification">
<p>Let \(\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 4 \\ 1 \\ 2 \end{array}} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 0 \end{array}} \right)\). Find \({\rm{B\hat AC}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><em><span style="font-family: times new roman,times; font-size: medium;">In this question, distance is in metres.</span></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Toy airplanes fly in a straight line at a constant speed. Airplane 1 passes through a point A.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Its position, <em>p</em> seconds after it has passed through A, is given by \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 4}\\<br>0<br>\end{array}} \right) + p\left( {\begin{array}{*{20}{c}}<br>{ - 2}\\<br>3\\<br>1<br>\end{array}} \right)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the coordinates of A.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the speed of the airplane in \({\text{m}}{{\text{s}}^{ - 1}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">After seven seconds the airplane passes through a point B.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the coordinates of B.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the distance the airplane has travelled during the seven seconds.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Airplane 2 passes through a point C. Its position <em>q</em> seconds after it passes </span><span style="font-family: times new roman,times; font-size: medium;">through C is given by \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 5}\\<br>8<br>\end{array}} \right) + q\left( {\begin{array}{*{20}{c}}<br>{ - 1}\\<br>2\\<br>a<br>\end{array}} \right),a \in \mathbb{R}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The angle between the flight paths of Airplane 1 and Airplane 2 is \({40^ \circ }\) . Find the </span><span style="font-family: times new roman,times; font-size: medium;">two values of <em>a</em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the points P(2, −1, 5) and Q(3, − 3, 8). Let \({L_1}\) be the line through P and Q.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(\overrightarrow {{\rm{PQ}}} = \left( {\begin{array}{*{20}{c}}<br>1\\<br>{ - 2}\\<br>3<br>\end{array}} \right)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The line \({L_1}\) may be represented by \({\boldsymbol{r}} = \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> { - 3} \\ <br> 8 <br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 2} \\ <br> 3 <br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) What information does the vector \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 3}\\<br>8<br>\end{array}} \right)\) give about \({L_1}\) ?</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Write down another vector representation for \({L_1}\) using \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 3}\\<br>8<br>\end{array}} \right)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The point \({\text{T}}( - 1{\text{, }}5{\text{, }}p)\) lies on \({L_1}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(p\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The point T also lies on \({L_2}\) with equation \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>{ - 3}\\<br>9\\<br>2<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>{ - 2}\\<br>q<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(q = - 3\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(\theta \) be the <strong>obtuse</strong> angle between \({L_1}\) and \({L_2}\) . Calculate the size of \(\theta \) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the points \({\text{A }}(1,{\text{ }}5,{\text{ }} - 7)\) and \({\text{B }}( - 9,{\text{ }}9,{\text{ }} - 6)\).</p>
</div>
<div class="specification">
<p class="p1">Let <span class="s1">C </span>be a point such that \(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 6 \\ { - 4} \\ 0 \end{array}} \right)\).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The line \(L\) </span>passes through B and is parallel to (AC)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\overrightarrow {{\text{AB}}} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of C.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down a vector equation for \(L\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(\left| {\overrightarrow {{\text{AB}}} } \right| = k\left| {\overrightarrow {{\text{AC}}} } \right|\), <span class="s1">find \(k\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The point D <span class="s1">lies on \(L\) </span>such that \(\left| {\overrightarrow {{\text{AB}}} } \right| = \left| {\overrightarrow {{\text{BD}}} } \right|\). Find the possible coordinates of D<span class="s1">.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The point O has coordinates (0 , 0 , 0) , point A has coordinates (1 , – 2 , 3) and point B has coordinates (– 3 , 4 , 2) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(\overrightarrow {{\rm{AB}}} = \left( {\begin{array}{*{20}{c}}<br>{ - 4}\\<br>6\\<br>{ - 1}<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({\rm{B}}\widehat {\rm{A}}{\rm{O}}\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The line \({L_1}\) has equation \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>{ - 3}\\<br>4\\<br>2<br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br>{ - 4}\\<br>6\\<br>{ - 1}<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Write down the coordinates of two points on \({L_1}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The line \({L_2}\) passes through A and is parallel to \(\overrightarrow {{\rm{OB}}} \) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find a vector equation for \({L_2}\) , giving your answer in the form \({\boldsymbol{r}} = {\boldsymbol{a}} + t{\boldsymbol{b}}\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Point \(C(k, - k,5)\) is on \({L_2}\) . Find the coordinates of C.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The line \({L_3}\) has equation \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 8}\\<br>0<br>\end{array}} \right) + p\left( {\begin{array}{*{20}{c}}<br>1\\<br>{ - 2}\\<br>{ - 1}<br>\end{array}} \right)\) and passes through the point C. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Find the value of <em>p</em> at C. </span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">Consider the lines \({L_1}\) and \({L_2}\) with equations \({L_1}\) : </span></span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\(\boldsymbol{r}=\left( \begin{array}{c}11\\8\\2\end{array} \right) + s\left( \begin{array}{c}4\\3\\ - 1\end{array} \right)\) and \({L_2}\) : </span></span></span><span style="background-color: #f7f7f7; line-height: normal;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\boldsymbol{r} = \left( \begin{array}{c}1\\1\\ - 7\end{array} \right) + t\left( \begin{array}{c}2\\1\\11\end{array} \right)\).</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The lines intersect at point \(\rm{P}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of \({\text{P}}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the lines are perpendicular.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The point \({\text{Q}}(7, 5, 3)\) lies on \({L_1}\). The point \({\text{R}}\) is the reflection of \({\text{Q}}\) in the line \({L_2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of \({\text{R}}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the cuboid (rectangular solid) OABCDEFG, where O is the origin, and \(\overrightarrow {{\rm{OA}}} = 4\boldsymbol{i}\) , \(\overrightarrow {{\rm{OC}}} = 3\boldsymbol{j}\) , \(\overrightarrow {{\rm{OD}}} = 2\boldsymbol{k}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/Lars.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(\overrightarrow {{\rm{OB}}} \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(\overrightarrow {{\rm{OF}}} \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Show that \(\overrightarrow {{\rm{AG}}} = - 4{\boldsymbol{i}} + 3{\boldsymbol{j}} + 2{\boldsymbol{k}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down a vector equation for</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the line OF;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the line AG.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the obtuse angle between the lines OF and AG.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the points A(\(5\), \(2\), \(1\)) , B(\(6\), \(5\), \(3\)) , and C(\(7\), \(6\), \(a + 1\)) , \(a \in{\mathbb{R}}\) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \({\rm{q}}\) be the angle between \(\overrightarrow {{\rm{AB}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">and \(\overrightarrow {{\rm{AC}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \(\overrightarrow {{\rm{AB}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \(\overrightarrow {{\rm{AC}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(a\) for which \({\rm{q}} = \frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">i. Show that \(\cos q = \frac{{2a + 14}}{{\sqrt {14{a^2} + 280} }}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">ii. Hence, find the value of a for which \({\rm{q}} = 1.2\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, find the value of a for which \({\rm{q}} = 1.2\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \({\boldsymbol{v}} = 3{\boldsymbol{i}} + 4{\boldsymbol{j}} + {\boldsymbol{k}}\) and \({\boldsymbol{w}} = {\boldsymbol{i}} + 2{\boldsymbol{j}} - 3{\boldsymbol{k}}\) . </span><span style="font-family: times new roman,times; font-size: medium;">The vector \({\boldsymbol{v}} + p{\boldsymbol{w}}\) is perpendicular to <strong><em>w</em></strong>. Find the value of <em>p</em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The points A and B <span class="s1">lie on a line \(L\)</span>, and have position vectors \(\left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ 2 \end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 1} \end{array}} \right)\) respectively. Let O <span class="s1">be the origin. This is shown on the following diagram.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-01_om_15.56.14.png" alt="M16/5/MATME/SP2/ENG/TZ1/10"></span></p>
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<div class="specification">
<p class="p1">The point C <span class="s1">also lies on \(L\)</span>, such that \(\overrightarrow {{\text{AC}}} = 2\overrightarrow {{\text{CB}}} \).</p>
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<div class="specification">
<p class="p1"><span class="s1">Let \(\theta \) </span>be the angle between \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{OC}}} \).</p>
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<div class="specification">
<p class="p1"><span class="s1">Let D be a point such that \(\overrightarrow {{\text{OD}}} = k\overrightarrow {{\text{OC}}} \)</span>, where \(k > 1\)<span class="s1">. Let E </span>be a point on \(L\) <span class="s1">such that \({\rm{C\hat ED}}\) </span>is a right angle. This is shown on the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_16.39.34.png" alt="M16/5/MATME/SP2/ENG/TZ1/10.d"></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\overrightarrow {{\text{AB}}} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ 0 \end{array}} \right)\).</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\theta \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\left| {\overrightarrow {{\text{DE}}} } \right| = (k - 1)\left| {\overrightarrow {{\text{OC}}} } \right|\sin \theta \).</p>
<p class="p2"><span class="s1">(ii) <span class="Apple-converted-space"> </span>The distance from D </span>to line \(L\) <span class="s1">is less than 3 </span>units. Find the possible values of \(k\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Line \({L_1}\) passes through points \({\text{A}}(1{\text{, }} - 1{\text{, }}4)\) and \({\text{B}}(2{\text{, }} - 2{\text{, }}5)\) .</span></p>
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<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Line \({L_2}\) has equation \({\boldsymbol{r}} = \left( \begin{array}{l}<br>2\\<br>4\\<br>7<br>\end{array} \right) + s\left( \begin{array}{l}<br>2\\<br>1\\<br>3<br>\end{array} \right)\) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\overrightarrow {{\rm{AB}}} \) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find an equation for \({L_1}\) in the form \({\boldsymbol{r}} = {\boldsymbol{a}} + t{\boldsymbol{b}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the angle between \({L_1}\) and \({L_2}\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The lines \({L_1}\) and \({L_2}\) intersect at point C. Find the coordinates of C.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram shows a parallelogram ABCD.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tired.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The coordinates of A, B and D are A(1, 2, 3) , B(6, 4,4 ) and D(2, 5, 5) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(\overrightarrow {{\rm{AB}}} = \left( {\begin{array}{*{20}{c}}<br>5\\<br>2\\<br>1<br>\end{array}} \right)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(\overrightarrow {{\rm{AD}}} \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) <strong>Hence</strong> show that \(\overrightarrow {{\rm{AC}}} = \left( {\begin{array}{*{20}{c}}<br>6\\<br>5\\<br>3<br>\end{array}} \right)\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the coordinates of point C.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} \).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong> find angle <em>A</em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, or otherwise, find the area of the parallelogram.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows two perpendicular vectors <strong><em>u </em></strong>and <strong><em>v</em></strong>.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_4.png" alt></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-size: medium; font-family: 'times new roman', times;">Let \(w = u - v\). Represent \(w\) on the diagram above.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span><span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">Given that \(u = \left( \begin{array}{c}3\\2\\1\end{array} \right)\) and </span></span></span></span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\(v = \left( \begin{array}{c}5\\n\\3\end{array} \right)\), where \(n \in \mathbb{Z}\), find \(</span></span><em style="font-family: 'times new roman', times; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">n\)</em><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">.</span></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the lines \({L_1}\) , \({L_2}\) , \({L_2}\) , and \({L_4}\) , with respective equations.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({L_1}\) : \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>1\\<br>2\\<br>3<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 2}\\<br>1<br>\end{array}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({L_2}\) : \(\left( \begin{array}{l}<br>x\\<br>y\\<br>z<br>\end{array} \right) = \left( \begin{array}{l}<br>1\\<br>2\\<br>3<br>\end{array} \right) + p\left( \begin{array}{l}<br>3\\<br>2\\<br>1<br>\end{array} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({L_3}\) : \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>0\\<br>1\\<br>0<br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br>{ - 1}\\<br>2\\<br>{ - a}<br>\end{array}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({L_4}\) : \(\left( {\begin{array}{*{20}{c}}<br>x\\<br>y\\<br>z<br>\end{array}} \right) = q\left( {\begin{array}{*{20}{c}}<br>{ - 6}\\<br>4\\<br>{ - 2}<br>\end{array}} \right)\)</span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the line that is parallel to \({L_4}\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the position vector of the point of intersection of \({L_1}\) and \({L_2}\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({L_1}\) is perpendicular to \({L_3}\) , find the value of <em>a</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p>Two points P and Q have coordinates (3, 2, 5) and (7, 4, 9) respectively.</p>
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<div class="specification">
<p>Let \({\mathop {{\text{PR}}}\limits^ \to }\) = 6<em><strong>i</strong></em> − <em><strong>j</strong></em> + 3<em><strong>k</strong></em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\mathop {{\text{PQ}}}\limits^ \to \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle between PQ and PR.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle PQR.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find the shortest distance from R to the line through P and Q.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The line <em>L<sub>1</sub></em> is represented by \({{\boldsymbol{r}}_1} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>5\\<br>3<br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br>1\\<br>2\\<br>3<br>\end{array}} \right)\) and the line <em>L<sub>2</sub></em> by \({{\boldsymbol{r}}_2} = \left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 3}\\<br>8<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>{ - 1}\\<br>3\\<br>{ - 4}<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> The lines <em>L<sub>1</sub></em> and <em>L<sub>2</sub></em> intersect at point T. Find the coordinates of T. </span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Line \({L_1}\) </span><span style="font-family: times new roman,times; font-size: medium;">has equation \({\boldsymbol{r}_1} = \left( {\begin{array}{*{20}{c}}<br>{10}\\<br>6\\<br>{ - 1}<br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 5}\\<br>{ - 2}<br>\end{array}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">and line \({L_2}\) has equation \({\boldsymbol{r}_2} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>1\\<br>{ - 3}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>3\\<br>5\\<br>2<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Lines \({L_1}\) and \({L_2}\) intersect at point A. Find the coordinates of A.</span></p>
</div>
<br><hr><br>