A Math

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H1 Math

H2 Math

IB HL Math

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IP Math

<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-paragraph">Find the shortest distance from the origin $O$ to the plane $p$ with equation $2x+3y+6z=10$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that point $A$ lies on plane $p$ and $OA=10$, find the acute angle between the plane $p$ and $OA$.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">Two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are such that&nbsp;$\left| \mathbf{a}\times \mathbf{b} \right|=\sqrt{3}$, $\left| \mathbf{a}-\mathbf{b} \right|=1$ and the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is ${{30}^{\circ }}$, find the possible values of $\left| \mathbf{a} \right|$ and $\left| \mathbf{b} \right|$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Two fixed points $C$ and $D$ have position vectors $\mathbf{c}$ and $\mathbf{d}$ respectively. By considering the result $\mathbf{c}\times \mathbf{c}=\mathbf{0}$, give a geometrical interpretation of the vector equation $\mathbf{r}\times \left( \mathbf{d}-\mathbf{c} \right)=\mathbf{c}\times \mathbf{d}$.<span class="inline-marks">[3]</span></div></li></ol>

Vectors - General

2023 NJC Promo Q5

Vectors

<p><strong>How to Prove the Formula of Distance between Two Parallel Planes</strong></p>

**How to Prove the Formula of Distance between Two Parallel Planes**

How to Prove the Formula of Distance between Two Parallel Planes

<p>The planes ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$ have equations $\mathbf{r}\cdot \left( \begin{matrix}1  \\ -3  \\  2  \\\end{matrix} \right)=-5$ and $\mathbf{r}\cdot \left( \begin{matrix}  2  \\-1  \\  3  \\\end{matrix} \right)=1$ respectively, and meet in the line ${{l}_{4}}$. Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the vector equation for ${{l}_{4}}$,</div></li><li><div class="list-item-paragraph">Find the acute angle between ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$.</div></li></ol>

The planes ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$ have equations $\mathbf{r}\cdot \left( \begin{matrix}1  \\ -3  \\  2  \\\end{matrix} \right)=-5$ and $\mathbf{r}\cdot \left( \begin{matrix}  2  \\-1  \\  3  \\\end{matrix} \right)=1$ respectively, and meet in the line ${{l}_{4}}$. Find

Unit 13.2 Worked Example 5

<p>Find the acute angle between direction vectors $\left( -6\mathbf{i}+\mathbf{j} \right)$ and $\left( \mathbf{i}-\mathbf{j}-8\mathbf{k} \right)$.</p>

Find the acute angle between direction vectors $\left( -6\mathbf{i}+\mathbf{j} \right)$ and $\left( \mathbf{i}-\mathbf{j}-8\mathbf{k} \right)$.

Unit 13.1 Worked Example 3

<ol class="i-list"><li><div class="list-item-paragraph">Find the exact shortest distance of the point $A\left( 3,1,-2 \right)$ to the plane ${{p}_{1}}:x-3y+2z=-5$.</div></li><li><div class="list-item-paragraph">Find the coordinates of foot of perpendicular of point $B\left( 7,4,-5 \right)$ to plane ${{p}_{1}}:x-3y+2z=-5$.</div></li></ol>

Unit 13.2 Worked Example 3

<p>The equation of a straight line $l$ is $\mathbf{r}=\left( \begin{matrix}
   1  \\
   2  \\
   3  \\
\end{matrix} \right)+t\left( \begin{matrix}
   -1  \\
   1  \\
   1  \\
\end{matrix} \right)$, where&nbsp;$t$ is a parameter. The point $A$on $l$ is given by $t=0$, and the origin of position vectors is $O$. <br></p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the acute angle between $OA$ and $\ell $, giving your answer correct to the nearest degree.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the position vector of the point $P$ on $\ell $ such that&nbsp;$OP$ is perpendicular to $\ell $.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">A point $Q$ on $\ell $ is such that the length of $OQ$ is $5$ units. Find the two possible position vectors of $Q$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The points $R$ and $S$ on $\ell $ are given by $t\,=\,\lambda $ and $t\,=\,2\lambda $ respectively. Show that there is no value of $\lambda $ for which $OR$ and $OS$ are perpendicular.<span class="inline-marks">[3]</span></div></li></ol>

The equation of a straight line $l$ is $\mathbf{r}=\left( \begin{matrix}
   1  \\
   2  \\
   3  \\
\end{matrix} \right)+t\left( \begin{matrix}
   -1  \\
   1  \\
   1  \\
\end{matrix} \right)$, where $t$ is a parameter. The point $A$on $l$ is given by $t=0$, and the origin of position vectors is $O$. 


N1993 P1 Q14

<ol class="i-list"><li><div class="list-item-paragraph">Find a vector equation of the line through the points $A$ and $B$ with position vectors $\mathrm{7}\mathbf{i}\,+\,8\mathbf{j}\,+\,9\mathbf{k}$ and $-\mathbf{i}-8\mathbf{j}+\mathbf{k}$ respectively.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The perpendicular to this line from the point $C$ with position vector $\mathbf{i}+8\mathbf{j}+3\mathbf{k}$ meets the line at point $N$. Find the position vector of $N$ and the ratio $AN:NB$.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Find the Cartesian equation of the line which is a reflection of the line $AC$ in the line $AB$.</div><p style="text-align: right;">[4]</p></li></ol>

N2012 P1 Q9

<p>In computer graphics rendering, ray tracing is the technique of tracing the paths of incident light rays onto surfaces and calculating the paths of their reflected lights rays to achieve a realistic image to a given point of view.</p>
<p>The point $A$ has coordinates $\left( 3,4,2 \right)$ and the line $l$ has equation $x-1=\frac{y+2}{3}=\frac{3-z}{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the cartesian equation of the plane $\Pi $ which contains $A$ and the line $l$ is $3x-y=5$.</div></li></ol>
<p>A light ray travels in a straight line, ${{l}_{SA}}$, from a point source at $S$ with coordinates $\left( 4,2,3 \right)$ towards point $A$ and is reflected off the surface of the plane $\Pi $. The reflected light ray travels along a line $m$ and the angles that lines ${{l}_{SA}}$ and $m$ make with the normal vector of the plane $\Pi $ are the same. Then line ${{l}_{SA}}$ and $m$ lie in a plane that is perpendicular to $\Pi $.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761054614716-2pkneluj13r.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEwNTQ2MTQ3MTYtMnBrbmVsdWoxM3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwOTIzOSwiZXhwIjoxNzg1NzA5NTM5fQ.5VtJJrXWlvdOtkfDYziApLQncDGDKavlMBWjuURuRsE" width="400" class="aligncenter" /></figure>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the foot of the perpendicular, $F$, from $S$ to the plane $\Pi $.</div></li><li><div class="list-item-paragraph">Hence by finding $\overrightarrow{FA}$, or otherwise, show that a cartesian equation for the line $m$ is $\frac{x-3}{2}=y-4=2-z$.</div></li><li><div class="list-item-paragraph">The viewing window in this digital simulation is the rectangle in the $xy$-plane starting at the origin with $4$ units length in the positive $x$-direction and $3$ units height in the positive $y$-direction. Determine whether the reflected light ray along line $m$ will reach the viewing window.</div></li></ol>

In computer graphics rendering, ray tracing is the technique of tracing the paths of incident light rays onto surfaces and calculating the paths of their reflected lights rays to achieve a realistic image to a given point of view.

The point $A$ has coordinates $\left( 3,4,2 \right)$ and the line $l$ has equation $x-1=\frac{y+2}{3}=\frac{3-z}{2}$.

A light ray travels in a straight line, ${{l}_{SA}}$, from a point source at $S$ with coordinates $\left( 4,2,3 \right)$ towards point $A$ and is reflected off the surface of the plane $\Pi $. The reflected light ray travels along a line $m$ and the angles that lines ${{l}_{SA}}$ and $m$ make with the normal vector of the plane $\Pi $ are the same. Then line ${{l}_{SA}}$ and $m$ lie in a plane that is perpendicular to $\Pi $.

2016 CJC CT Q5

<p>A vector which has a magnitude (length) of ${1.}$ ${\Rightarrow \left| \hat{\mathbf{a}} \right| = 1}$<br>Unit vector of ${\mathbf{a}}$ , ${\hat{\mathbf{a}} = \frac{\mathbf{a}}{\left| \mathbf{a} \right|}}$</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768631506934-h7p5fztz7bj.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg2MzE1MDY5MzQtaDdwNWZ6dHo3YmoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwOTIzOSwiZXhwIjoxNzg1NzA5NTM5fQ.AbR5YQD6sMiUTrA-O4MZnFL1RAxrzUs8XZQBH-dHccg" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

A vector which has a magnitude (length) of ${1.}$ ${\Rightarrow \left| \hat{\mathbf{a}} \right| = 1}$
Unit vector of ${\mathbf{a}}$ , ${\hat{\mathbf{a}} = \frac{\mathbf{a}}{\left| \mathbf{a} \right|}}$

Unit Vector 1

<p>The plane $p$ has equation $\mathbf{r}=\left( \begin{matrix} 1 \\ 0 \\ -3 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 1 \\ 2 \\ 0 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 2 \\ 3 \\ -4 \\ \end{matrix} \right)$, and the line $l$ has equation $\mathbf{r}=\left( \begin{matrix} -10 \\ 0 \\ 4 \\ \end{matrix} \right)+t\left( \begin{matrix} 8 \\ -4 \\ 1 \\ \end{matrix} \right)$, where $\lambda $, $\mu $ and $t$ are parameters.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $l$ is perpendicular to $p$ and find the values of $\lambda $, $\mu $ and $t$ which give the coordinates of the point at which $l$ and $p$ intersect.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Find the cartesian equations of the planes such that the perpendicular distance from each plane to $p$ is $2$.</div><p style="text-align: right;">[5]</p></li></ol>

The plane $p$ has equation $\mathbf{r}=\left( \begin{matrix} 1 \\ 0 \\ -3 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 1 \\ 2 \\ 0 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 2 \\ 3 \\ -4 \\ \end{matrix} \right)$, and the line $l$ has equation $\mathbf{r}=\left( \begin{matrix} -10 \\ 0 \\ 4 \\ \end{matrix} \right)+t\left( \begin{matrix} 8 \\ -4 \\ 1 \\ \end{matrix} \right)$, where $\lambda $, $\mu $ and $t$ are parameters.

2019 JPJC P2 Q5

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761365257856-bwiph4eca4.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjUyNTc4NTYtYndpcGg0ZWNhNC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzA5MjM5LCJleHAiOjE3ODU3MDk1Mzl9.qGC8J5_r_s6M6FCM6bMFtgDSbWmc9kH9rap1wWLl-5M" style="max-width: 300px; height: auto; display: block;" class="aligncenter" /></figure>
<p>The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.</p>
<p>The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Give a reason why $p=0$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Write down a vector equation of the line passing through the points $D$ and $E$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that $q=2$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, find the obtuse angle between the walls $OADE$ and $DEFG$.<span class="inline-marks">[3]</span></div></li></ol>
<p>A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">Show that the climber is on the plane $DEFG$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.</p>
<ol class="abc-list" start="6"><li><div class="list-item-paragraph">Find the exact shortest distance from the climber to $l$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.

The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.

A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.

The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.

2022 YIJC Promo Q11

<p>The plane ${{\Pi }_{1}}$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   \alpha   \\
   \beta   \\
   0  \\
\end{matrix}  \\
\end{matrix} \right)=30$, where&nbsp;$\alpha $ and $\beta $ are positive constants, and contains the point $A$ with coordinates $\left( 0,\ 10,\ 5 \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that the perpendicular distance from the origin $O$ to the plane ${{\Pi }_{1}}$ is 6 units, find $\alpha $ and $\beta $.<span class="inline-marks">[5]</span></div></li></ol>
<p>Another plane ${{\Pi }_{2}}$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   1  \\
   -1  \\
   -2  \\
\end{matrix}  \\
\end{matrix} \right)=4$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the acute angle between the line $OA$ and the plane ${{\Pi }_{2}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find a Cartesian equation of the plane ${{\Pi }_{3}}$ which contains the line $OA$ and is perpendicular to the plane ${{\Pi }_{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

The plane ${{\Pi }_{1}}$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   \alpha   \\
   \beta   \\
   0  \\
\end{matrix}  \\
\end{matrix} \right)=30$, where $\alpha $ and $\beta $ are positive constants, and contains the point $A$ with coordinates $\left( 0,\ 10,\ 5 \right)$.

Another plane ${{\Pi }_{2}}$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   1  \\
   -1  \\
   -2  \\
\end{matrix}  \\
\end{matrix} \right)=4$.

2023 JPJC Promo Q11

<p>The plane ${{p}_{1}}$ contains points $A$ and $B$, with position vectors $\left( \begin{matrix}2 \\0 \\1 \\\end{matrix} \right)$ and $\left( \begin{matrix}8 \\-6 \\3 \\\end{matrix} \right)$ respectively, and is parallel to the line with direction vector $\left( \begin{matrix}7 \\-5 \\3 \\\end{matrix} \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find a cartesian equation of the plane ${{p}_{1}}$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The plane ${{p}_{2}}$ has equation $\mathbf{r}\cdot (\mathbf{i}-2\mathbf{j}+\mathbf{k})=3$. The planes ${{p}_{1}}$ and ${{p}_{2}}$ meet in the line $l$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find a vector equation of the line $l$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the acute angle between the planes ${{p}_{1}}$ and ${{p}_{2}}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>It is given that the point $R$ with position vector $-17\mathbf{i}-5\mathbf{j}+10\mathbf{k}$ lies on ${{p}_{2}}$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the foot of perpendicular from the point $R$ to the plane ${{p}_{1}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence, find a vector equation of the plane ${{p}_{3}}$ which is a reflection of the plane ${{p}_{2}}$ in the plane ${{p}_{1}}$ in the form $\mathbf{r}\cdot \mathbf{n}=d$, where $d$ is a constant to be determined.</div><p style="text-align: right;">[4]</p></li></ol>

The plane ${{p}_{1}}$ contains points $A$ and $B$, with position vectors $\left( \begin{matrix}2 \\0 \\1 \\\end{matrix} \right)$ and $\left( \begin{matrix}8 \\-6 \\3 \\\end{matrix} \right)$ respectively, and is parallel to the line with direction vector $\left( \begin{matrix}7 \\-5 \\3 \\\end{matrix} \right)$.

The plane ${{p}_{2}}$ has equation $\mathbf{r}\cdot (\mathbf{i}-2\mathbf{j}+\mathbf{k})=3$. The planes ${{p}_{1}}$ and ${{p}_{2}}$ meet in the line $l$.

It is given that the point $R$ with position vector $-17\mathbf{i}-5\mathbf{j}+10\mathbf{k}$ lies on ${{p}_{2}}$.

2017 CJC J1 MYE Q8

<p>The position vectors of two points $A$ and $B$ are $\left( 3\mathbf{i}-\mathbf{j}+4\mathbf{k} \right)$ and $\left( 2\mathbf{i}+\mathbf{j}+2\mathbf{k} \right)$ respectively. Points $A$, $B$ and $C$ are collinear. Given that point $C$ is $6$ units away from $A$ , find the possible position vectors of $C$.</p>

The position vectors of two points $A$ and $B$ are $\left( 3\mathbf{i}-\mathbf{j}+4\mathbf{k} \right)$ and $\left( 2\mathbf{i}+\mathbf{j}+2\mathbf{k} \right)$ respectively. Points $A$, $B$ and $C$ are collinear. Given that point $C$ is $6$ units away from $A$ , find the possible position vectors of $C$.

Unit 13.1 Worked Example 1

<p>Referred to the origin $O$, the position vectors of the points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively, and $OAB$ is a triangle.</p>
<p>Show that the area of the triangle is given by $\frac{1}{2}\left| \mathbf{a}\times \mathbf{b} \right|$.<span class="inline-marks">[2]</span></p>
<p>Show that the maximum area of the triangle is $\frac{1}{2}\left| \mathbf{a} \right|\left| \mathbf{b} \right|$.<span class="inline-marks">[2]</span></p>
<p>By considering $\left| \mathbf{a}\centerdot \mathbf{b} \right|$ where&nbsp;$\mathbf{a}=\left( \begin{matrix}
{{a}_{1}} \\
{{a}_{2}} \\
{{a}_{3}} \\
\end{matrix} \right)$ and $\mathbf{b}=\left( \begin{matrix}
{{b}_{1}} \\
{{b}_{2}} \\
{{b}_{3}} \\
\end{matrix} \right)$, show that</p>
<p style="text-align: center">${{\left( {{a}_{1}}{{b}_{1}}+{{a}_{2}}{{b}_{2}}+{{a}_{3}}{{b}_{3}} \right)}^{2}}\le \left( {{a}_{1}}^{2}+{{a}_{2}}^{2}+{{a}_{3}}^{2} \right)\left( {{b}_{1}}^{2}+{{b}_{2}}^{2}+{{b}_{3}}^{2} \right)$<span class="inline-marks">[3]</span></p>

Referred to the origin $O$, the position vectors of the points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively, and $OAB$ is a triangle.

Show that the area of the triangle is given by $\frac{1}{2}\left| \mathbf{a}\times \mathbf{b} \right|$.

Show that the maximum area of the triangle is $\frac{1}{2}\left| \mathbf{a} \right|\left| \mathbf{b} \right|$.

By considering $\left| \mathbf{a}\centerdot \mathbf{b} \right|$ where $\mathbf{a}=\left( \begin{matrix}
{{a}_{1}} \\
{{a}_{2}} \\
{{a}_{3}} \\
\end{matrix} \right)$ and $\mathbf{b}=\left( \begin{matrix}
{{b}_{1}} \\
{{b}_{2}} \\
{{b}_{3}} \\
\end{matrix} \right)$, show that

${{\left( {{a}_{1}}{{b}_{1}}+{{a}_{2}}{{b}_{2}}+{{a}_{3}}{{b}_{3}} \right)}^{2}}\le \left( {{a}_{1}}^{2}+{{a}_{2}}^{2}+{{a}_{3}}^{2} \right)\left( {{b}_{1}}^{2}+{{b}_{2}}^{2}+{{b}_{3}}^{2} \right)$

2013 MI P1 Q11

<p>Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel vectors. Point $C$ lies on $OA$, between $O$ and $A$, such that $OC:CA=2:1$. Point $D$ lies on $OB$ produced such that $OD:BD=3:2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the position vectors $\overrightarrow{OC}$ and $\overrightarrow{OD}$, giving your answers in terms of $\mathbf{a}$ and $\mathbf{b}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the point $E$ where the lines $BC$ and $AD$ meet has position vector $\frac{4}{3}\mathbf{a}-\mathbf{b}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Show that the area of triangle $CDE$ can be written as $k\left| \mathbf{a}\times \mathbf{b} \right|$, where $k$ is a constant to be found.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">It is given that the point $F$ is on $BO$ produced, and $OE$ bisects the angle $AOF$. Find the ration $OA:OB$.</div><p style="text-align: right;">[3]</p></li></ol>

Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel vectors. Point $C$ lies on $OA$, between $O$ and $A$, such that $OC:CA=2:1$. Point $D$ lies on $OB$ produced such that $OD:BD=3:2$.

2020 RI P2 Q5

<p>The equation of line ${{l}_{1}}$ is given by $\mathbf{r}=\left( \begin{matrix} -12  \\-22  \\11  \\\end{matrix} \right)+\lambda \left( \begin{matrix}  4  \\ 4  \\  -7  \\\end{matrix} \right);\lambda \in \mathbb{R}$ and the point $A$ has coordinates $\left( 3,1,-2 \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Exact shortest distance of $A$ to ${{l}_{1}}$,</div></li><li><div class="list-item-paragraph">Foot of perpendicular of $A$ to ${{l}_{1}}$,</div></li><li><div class="list-item-paragraph">Mirror image of point $A$ in the line ${{l}_{1}}$.</div></li></ol>

The equation of line ${{l}_{1}}$ is given by $\mathbf{r}=\left( \begin{matrix} -12  \\-22  \\11  \\\end{matrix} \right)+\lambda \left( \begin{matrix}  4  \\ 4  \\  -7  \\\end{matrix} \right);\lambda \in \mathbb{R}$ and the point $A$ has coordinates $\left( 3,1,-2 \right)$.

Unit 13.2 Worked Example 1

<p>The line $l$ passes through the points $A$ and $B$ with coordinates $\left( \mathrm{5, 2, 4} \right)$ and $\left( 4,\,-1,\,3 \right)$respectively. The plane $p$ has equation $4x+7y+5z=24$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The point $C$ lies on $l$ such that the foot of perpendicular of $C$ onto $p$ has coordinates $\left( 3,\mathrm{ }1,\mathrm{ }1 \right)$. Find the coordinates of $C$.</div><p style="text-align: right;">[4]</p></li></ol>
<p>Plane ${{p}_{1}}$ has equation $3x-2y+\lambda z=\mu $.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">What can be said about the values of $\lambda $ and $\mu $ if $l$ does not intersect ${{p}_{1}}$?</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence find the exact values of $\mu $ if the distance between ${{p}_{1}}$ and $l$ is $2$ units.</div><p style="text-align: right;">[3]</p></li></ol>

The line $l$ passes through the points $A$ and $B$ with coordinates $\left( \mathrm{5, 2, 4} \right)$ and $\left( 4,\,-1,\,3 \right)$respectively. The plane $p$ has equation $4x+7y+5z=24$.

Plane ${{p}_{1}}$ has equation $3x-2y+\lambda z=\mu $.

2018 NYJC P1 Q5

<p>Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{j}+3\mathbf{k}$, find:</p>
<ol class="i-list"><li><div class="list-item-paragraph">a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">the exact value of $\sin \theta $, where $\theta $ is the angle between $\mathbf{a}$ and $\mathbf{b}$.</div></li></ol>

Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{j}+3\mathbf{k}$, find:

Unit 13.1 Worked Example 4

<p>The plane ${{\prod }_{1}}$ passes through $\left( 3,-1,2 \right)$ and is perpendicular to the line $\mathbf{r}=2\mathbf{i}+4\mathbf{j}-3\mathbf{k}+\lambda \left( 5\mathbf{i}+2\mathbf{j}-\mathbf{k} \right)$. The plane ${{\prod }_{2}}$ contains the points $\left( 2,3,2 \right)$, $\left( 4,1,-1 \right)$ and $\left( 0,-1,2 \right)$.</p>
<ol class="i-list"><li data-list-item-id="item-1761038051502-ny93y0n2o" data-list-item-path="item-1761038051502-ny93y0n2o"><div class="list-item-paragraph">Show that the acute angle, $\theta $, between the planes ${{\prod }_{1}}$ and ${{\prod }_{2}}$ is such that&nbsp;$\cos \theta =\frac{\sqrt{30}}{15}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761038052065-h25aq80j2" data-list-item-path="item-1761038052065-h25aq80j2"><div class="list-item-paragraph">Show that the line of intersection, $L$, of the planes ${{\prod }_{1}}$ and ${{\prod }_{2}}$ has vector equation $\mathbf{r}=\left( \begin{matrix}0 \\9 \\7 \end{matrix} \right)+\mu \left(\begin{matrix}-1 \\4 \\3 \end{matrix} \right)$, $\mu \in \mathbb{R}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The plane ${{\prod }_{3}}$ has the equation $4\left( k-2 \right)x+\left( k+1 \right)y-4{{k}^{2}}z=8$, where&nbsp;$k$ is a constant.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1761038070815-wjth4dt0p" data-list-item-path="item-1761038070815-wjth4dt0p"><div class="list-item-paragraph">The three planes ${{\prod }_{1}}$,${{\prod }_{2}}$ and ${{\prod }_{3}}$have no points in common. By considering the relationship between the line $L$ and the plane ${{\prod }_{3}}$, find the possible values of $k$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761038071324-cfjtcobex" data-list-item-path="item-1761038071324-cfjtcobex"><div class="list-item-paragraph">For the positive value of $k$ found in <strong>(iii)</strong>, find the distance between $L$ and ${{\prod }_{3}}$.<span class="inline-marks">[2]</span></div></li></ol>

The plane ${{\prod }_{1}}$ passes through $\left( 3,-1,2 \right)$ and is perpendicular to the line $\mathbf{r}=2\mathbf{i}+4\mathbf{j}-3\mathbf{k}+\lambda \left( 5\mathbf{i}+2\mathbf{j}-\mathbf{k} \right)$. The plane ${{\prod }_{2}}$ contains the points $\left( 2,3,2 \right)$, $\left( 4,1,-1 \right)$ and $\left( 0,-1,2 \right)$.

The plane ${{\prod }_{3}}$ has the equation $4\left( k-2 \right)x+\left( k+1 \right)y-4{{k}^{2}}z=8$, where $k$ is a constant.

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