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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The lines $l_1$ and $l_2$ have equations \[\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\3\end{pmatrix}\qquad\text{and}\qquad\mathbf{r}=\begin{pmatrix}3\\4\\p\end{pmatrix}+\mu\begin{pmatrix}2\\1\\q\end{pmatrix}\] respectively, where&nbsp;$p$ and $q$ are constants, and $\lambda$ and $\mu$ are parameters.</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q11-item01" data-list-item-path="nyjc-q11-item01"><div class="list-item-paragraph">Suppose that&nbsp;$l_1$ intersects $l_2$.</div><ol class="i-list"><li data-list-item-id="nyjc-q11-item01-sub-item01" data-list-item-path="nyjc-q11-item01/nyjc-q11-item01-sub-item01"><div class="list-item-paragraph">Find the coordinates of the point of intersection and state a linear relationship between $p$ and $q$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="nyjc-q11-item01-sub-item02" data-list-item-path="nyjc-q11-item01/nyjc-q11-item01-sub-item02"><div class="list-item-paragraph">Find the acute angle between $l_1$ and $l_2$ when&nbsp;$q=2$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="nyjc-q11-item02" data-list-item-path="nyjc-q11-item02"><div class="list-item-paragraph">By using vector product, show that a vector perpendicular to both&nbsp;$l_1$ and $l_2$ is $\begin{pmatrix}2q+3\\q-6\\m\end{pmatrix}$, where&nbsp;$m$ is a constant to be determined.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q11-item03" data-list-item-path="nyjc-q11-item03"><div class="list-item-paragraph">The plane $\pi$ contains $l_2$ and is parallel to $l_1$. Find the shortest distance between $l_1$ and $\pi$ in terms of $p$ and $q$.<span class="inline-marks">[4]</span></div></li></ol>

The lines $l_1$ and $l_2$ have equations \[\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\3\end{pmatrix}\qquad\text{and}\qquad\mathbf{r}=\begin{pmatrix}3\\4\\p\end{pmatrix}+\mu\begin{pmatrix}2\\1\\q\end{pmatrix}\] respectively, where $p$ and $q$ are constants, and $\lambda$ and $\mu$ are parameters.

By using vector product, show that a vector perpendicular to both $l_1$ and $l_2$ is $\begin{pmatrix}2q+3\\q-6\\m\end{pmatrix}$, where $m$ is a constant to be determined.

The plane $\pi$ contains $l_2$ and is parallel to $l_1$. Find the shortest distance between $l_1$ and $\pi$ in terms of $p$ and $q$.

Distance from a Point to a Plane

Angles Between Two Lines

Intersection of Two Lines

Vector (Cross) Product

Equations of Planes

2026 NYJC P1 Q11

Vectors

<ol class="abc-list"><li data-list-item-id="h2-2025-ejc-promo-q3-ejc-q3-item-block-0-item-0" data-list-item-path="h2-2025-ejc-promo-q3-ejc-q3-item-block-0-item-0"><div class="list-item-paragraph">Given that&nbsp;$\mathbf{a}\times\mathbf{b}=\mathbf{0}$, what can be deduced about the vectors $\mathbf{a}$ and $\mathbf{b}$?<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q3-ejc-q3-item-block-0-item-1" data-list-item-path="h2-2025-ejc-promo-q3-ejc-q3-item-block-0-item-1"><div class="list-item-paragraph">Find a unit vector $\mathbf{n}$ such that&nbsp;$\mathbf{n}\times(2\mathbf{i}-\mathbf{j}+3\mathbf{k})=\mathbf{0}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q3-ejc-q3-item-block-0-item-2" data-list-item-path="h2-2025-ejc-promo-q3-ejc-q3-item-block-0-item-2"><div class="list-item-paragraph">Find the cosine of the acute angle between $2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and the $y$-axis.<span class="inline-marks">[2]</span></div></li></ol>

Given that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$, what can be deduced about the vectors $\mathbf{a}$ and $\mathbf{b}$?

Find a unit vector $\mathbf{n}$ such that $\mathbf{n}\times(2\mathbf{i}-\mathbf{j}+3\mathbf{k})=\mathbf{0}$.

Find the cosine of the acute angle between $2\mathbf{i}-\mathbf{j}+3\mathbf{k}$ and the $y$-axis.

Parallel Vectors

Unit Vectors

2025 EJC Promo Q3

<p>With reference to the point $O$ as the origin and the $x$-$y$ plane as a horizontal plane, the pyramid $OPQRV$ has a parallelogram base $OPQR$ and height $OV$. The position vectors of the points $P$ and $R$ are $-3\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $5\mathbf{i} - 2\mathbf{j} - \mathbf{k}$ respectively.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the coordinates of the point $S$ that lies on the line $PR$ such that the distance from $O$ to $S$ is a minimum.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the cartesian equations of the planes such that the perpendicular distance from each plane to the base $OPQR$ is $2\sqrt{86}$ units.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acute angle between $OV$ and the vertical.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Given that&nbsp;$\overrightarrow{QV}$ is parallel to the vector $-5\mathbf{j} + 8\mathbf{k}$, find the position vector of the point $V$. Hence find the exact volume of the pyramid $OPQRV$.<span class="inline-marks">[5]</span></div></li></ol>
<p>$\text{[Volume of a pyramid} = \frac{1}{3} \times \text{base area} \times \text{height]}$</p>

With reference to the point $O$ as the origin and the $x$-$y$ plane as a horizontal plane, the pyramid $OPQRV$ has a parallelogram base $OPQR$ and height $OV$. The position vectors of the points $P$ and $R$ are $-3\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $5\mathbf{i} - 2\mathbf{j} - \mathbf{k}$ respectively.

$\text{[Volume of a pyramid} = \frac{1}{3} \times \text{base area} \times \text{height]}$

Perpendicular Projection and Reflection in a Line

Three-Dimensional Geometry

Vectors in a Parallelogram

Areas of Triangles and Parallelograms Using Vectors

2025 VJC P2 Q3

<p>The line $l_1$ has equation $x - 1 = 2 - y = -z$.</p>
<p>The line $l_2$ passes through the point $A(-2, 1, 7)$ and is parallel to the vector $-3\mathbf{i} + \mathbf{j} + 5\mathbf{k}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down vector equations for&nbsp;$l_1$ and $l_2$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the lines $l_1$ and $l_2$ intersect and find the coordinates of $E$, the point of intersection of $l_1$ and $l_2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The acute angle between the lines $l_1$ and $l_2$ is denoted by $\theta$. By finding $\cos \theta$, show that \[\sin \theta = \frac{2\sqrt{2}}{\sqrt{35}}.\] Hence find the shortest distance from $A$ to the line $l_1$, leaving your answer in exact form.</span><span class="inline-marks">[5]</span></div></li></ol>

The line $l_1$ has equation $x - 1 = 2 - y = -z$.

The line $l_2$ passes through the point $A(-2, 1, 7)$ and is parallel to the vector $-3\mathbf{i} + \mathbf{j} + 5\mathbf{k}$.

Write down vector equations for $l_1$ and $l_2$.

Show that the lines $l_1$ and $l_2$ intersect and find the coordinates of $E$, the point of intersection of $l_1$ and $l_2$.

The acute angle between the lines $l_1$ and $l_2$ is denoted by $\theta$. By finding $\cos \theta$, show that \[\sin \theta = \frac{2\sqrt{2}}{\sqrt{35}}.\] Hence find the shortest distance from $A$ to the line $l_1$, leaving your answer in exact form.

Equations of Lines

Distance from a Point to a Line

<p>The lines $l_1$ and $l_2$ have equations</p>
<p style="text-align: center">\[\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + s \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} \text{ and}\hspace{0.5em} \mathbf{r} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ b \\ 0 \end{pmatrix}\]</p>
<p>respectively, where&nbsp;$b$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">For the case where the lines intersect, calculate the acute angle between them.</span></div></li><li><div class="list-item-content"><span class="list-item-text">For the case when&nbsp;$b = 1$, find the position vectors of the points $P$ on $l_1$ and $Q$ on $l_2$ such that&nbsp;$O$, $P$ and $Q$ are collinear, where&nbsp;$O$ is the origin.</span></div></li></ol>

The lines $l_1$ and $l_2$ have equations

\[\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + s \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} \text{ and } \mathbf{r} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ b \\ 0 \end{pmatrix}\]

For the case where the lines intersect, calculate the acute angle between them.

For the case when $b = 1$, find the position vectors of the points $P$ on $l_1$ and $Q$ on $l_2$ such that $O$, $P$ and $Q$ are collinear, where $O$ is the origin.

Collinearity Using Vectors

N2005 P1 Q14

<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/1775962953671-l5kobtsrm5n.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU5NjI5NTM2NzEtbDVrb2J0c3JtNW4uanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwODYzMSwiZXhwIjoxNzkzNTA4OTMxfQ.K_sV5Ttgi0TPIqBXspZVp98-aeQEprljtgdQ6Mhcpd0" style="max-width: min(270px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>In the diagram, $O$ is the centre of the rectangular base $ABCD$ of a right pyramid with vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $BC$ and $OV$ respectively. The length of $AB$, $BC$ and $OV$ are $12\text{ cm}$, $6\text{ cm}$ and $6\text{ cm}$ respectively. A line $l$ has Cartesian equation $\frac{-x-4}{10} = y+2 = \frac{t-z}{-2}.$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of $AV$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the line $l$ intersects $AV$ at $M$, find the position vector of $M$ and the value of $t$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between $AV$ and the line $l$. Hence find the perpendicular distance from $A$ to the line $l$.</span></div></li></ol>

In the diagram, $O$ is the centre of the rectangular base $ABCD$ of a right pyramid with vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $BC$ and $OV$ respectively. The length of $AB$, $BC$ and $OV$ are $12\text{ cm}$, $6\text{ cm}$ and $6\text{ cm}$ respectively. A line $l$ has Cartesian equation $\frac{-x-4}{10} = y+2 = \frac{t-z}{-2}.$

If the line $l$ intersects $AV$ at $M$, find the position vector of $M$ and the value of $t$.

Find the acute angle between $AV$ and the line $l$. Hence find the perpendicular distance from $A$ to the line $l$.

2006 HCI P1 Q14

<p>A line $l$ passes through the points $A$ and $B$ with coordinates $(0, -2, 2)$ and $(1, 0, 1)$ respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between $\overrightarrow{OA}$ and the line $l$, where&nbsp;$O$ is the origin.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the shortest distance from $O$ to the line $l$, leaving your answer in exact form.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>A plane $\pi_1$ has equation $\mathbf{r} \cdot (2\mathbf{i} - \mathbf{j}) = 2$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Show that the line $l$ lies in the plane $\pi_1$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>A second plane $\pi_2$ contains the line $l$ and is perpendicular to the plane $\pi_1$.</p>
<ol class="i-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the plane $\pi_2$.</span><span class="inline-marks">[2]</span></div></li></ol>

A line $l$ passes through the points $A$ and $B$ with coordinates $(0, -2, 2)$ and $(1, 0, 1)$ respectively.

Find the acute angle between $\overrightarrow{OA}$ and the line $l$, where $O$ is the origin.

Hence, find the shortest distance from $O$ to the line $l$, leaving your answer in exact form.

A plane $\pi_1$ has equation $\mathbf{r} \cdot (2\mathbf{i} - \mathbf{j}) = 2$.

Show that the line $l$ lies in the plane $\pi_1$.

A second plane $\pi_2$ contains the line $l$ and is perpendicular to the plane $\pi_1$.

Find the cartesian equation of the plane $\pi_2$.

Relationships Between a Line and a Plane

2015 MI P1 Q7

<p>With reference to the origin $O$, the points $A$, $P$ and $Q$ have position vectors $\mathbf{a}$, $\mathbf{p}$ and $\mathbf{q}$ respectively. A straight line $l$ through the point $A$ is parallel to a unit vector $\mathbf{e}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The point $Q$ lies on $l$. Show that&nbsp;$(\mathbf{q} - \mathbf{a}) \times \mathbf{e} = \mathbf{0}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$P$ is not on $l$, give the geometrical meaning of $|(\mathbf{p} - \mathbf{a}) \times \mathbf{e}|$.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>It is also given that&nbsp;$\mathbf{p} = 3\mathbf{q}$ and $AQ$ is $2$ units.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Using the results in <strong>(a)</strong> and <strong>(b)</strong>, or otherwise, show that the area of triangle $APQ$ can be written as&nbsp;$k|\mathbf{q} \times \mathbf{e}|$, where&nbsp;$k$ is a constant to be determined.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, find the acute angle between $l$ and $PQ$, given further that the area of triangle $\Delta APQ$ is $3$ units${}^2$ and $|\mathbf{q}| = 2$.</span><span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin $O$, the points $A$, $P$ and $Q$ have position vectors $\mathbf{a}$, $\mathbf{p}$ and $\mathbf{q}$ respectively. A straight line $l$ through the point $A$ is parallel to a unit vector $\mathbf{e}$.

Given that $P$ is not on $l$, give the geometrical meaning of $|(\mathbf{p} - \mathbf{a}) \times \mathbf{e}|$.

It is also given that $\mathbf{p} = 3\mathbf{q}$ and $AQ$ is $2$ units.

Hence, or otherwise, find the acute angle between $l$ and $PQ$, given further that the area of triangle $\Delta APQ$ is $3$ units${}^2$ and $|\mathbf{q}| = 2$.

2025 JPJC P2 Q2

<p>Relative to the origin $O$, point $A$ has position vector given by $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. The line $l_1$ passes through the point $A$ and is parallel to $\begin{pmatrix} t \\ 3 - t^2 \\ 1 \end{pmatrix}$. The plane $p$ has equation $\mathbf{r} = \lambda \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 0 \\ t \end{pmatrix}, \lambda, \mu \in \mathbb{R}$ where&nbsp;$t$ is a real constant. It is known that&nbsp;$l_1$ and $p$ are parallel and $l_1$ is not on $p$.</p>
<ol class="abc-list"><li data-list-item-id="item-1768711344529-8nihmues6" data-list-item-path="item-1768711344529-8nihmues6"><div class="list-item-paragraph">Show that&nbsp;$t = -1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1768711345279-s547r63ey" data-list-item-path="item-1768711345279-s547r63ey"><div class="list-item-paragraph">The line $l_2$ passes through the origin and point $A$. Find the acute angle between $l_1$ and $l_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1768711345923-79fmat0do" data-list-item-path="item-1768711345923-79fmat0do"><div class="list-item-paragraph">The vector $\mathbf{n}$ is a unit vector normal to $p$. State the geometrical meaning of $|\mathbf{a} \cdot \mathbf{n}|$ and find the exact value of $|\mathbf{a} \cdot \mathbf{n}|$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1768711346929-qm8zctapw" data-list-item-path="item-1768711346929-qm8zctapw"><div class="list-item-paragraph">Find a vector equation of the line of reflection of $l_2$ in $p$.<span class="inline-marks">[4]</span></div></li></ol>

Relative to the origin $O$, point $A$ has position vector given by $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. The line $l_1$ passes through the point $A$ and is parallel to $\begin{pmatrix} t \\ 3 - t^2 \\ 1 \end{pmatrix}$. The plane $p$ has equation $\mathbf{r} = \lambda \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 0 \\ t \end{pmatrix}, \lambda, \mu \in \mathbb{R}$ where $t$ is a real constant. It is known that $l_1$ and $p$ are parallel and $l_1$ is not on $p$.

Perpendicular Projection and Reflection in a Plane

Projection Lengths

2025 EJC P2 Q4

<p>The line ${{l}_{2}}$ passes through $A\left( 3,1,-2 \right)$ and is parallel to the direction vector $\left( \begin{matrix}7  \\15  \\1  \\\end{matrix} \right)$.

${{l}_{2}}$ intersects ${{l}_{1}}:\mathbf{r}=\left( \begin{matrix}  -12  \\ -22  \\ 11  \\\end{matrix} \right)+\lambda \left( \begin{matrix}4  \\ 4  \\ -7  \\\end{matrix} \right);\lambda \in \mathbb{R}$ at $C$.Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the coordinates of $C$,</div></li><li><div class="list-item-paragraph">the acute angle between ${{l}_{1}}$ and ${{l}_{2}}$,</div></li><li><div class="list-item-paragraph">the vector line equation of mirror image of ${{l}_{2}}$ in ${{l}_{1}}$.</div></li></ol>

The line ${{l}_{2}}$ passes through $A\left( 3,1,-2 \right)$ and is parallel to the direction vector $\left( \begin{matrix}7  \\15  \\1  \\\end{matrix} \right)$.

${{l}_{2}}$ intersects ${{l}_{1}}:\mathbf{r}=\left( \begin{matrix}  -12  \\ -22  \\ 11  \\\end{matrix} \right)+\lambda \left( \begin{matrix}4  \\ 4  \\ -7  \\\end{matrix} \right);\lambda \in \mathbb{R}$ at $C$.Find

Unit 13.2 Worked Example 2

<p>Determine whether the following pair of lines are parallel, intersecting or skew. Find the position vector of the intersection point (if any) and the angle between the lines.</p>
<p>${{l}_{1}}\mathrm{:r}=\left( \begin{matrix} 3 \\1\\ -5 \\\end{matrix} \right)+\lambda \left( \begin{matrix} -1\\ -2\\ 3 \\ \end{matrix} \right)$, ${{l}_{2}}\mathrm{:}x+3=1-y,z=0$</p>

Determine whether the following pair of lines are parallel, intersecting or skew. Find the position vector of the intersection point (if any) and the angle between the lines.

${{l}_{1}}\mathrm{:r}=\left( \begin{matrix} 3 \\1\\ -5 \\\end{matrix} \right)+\lambda \left( \begin{matrix} -1\\ -2\\ 3 \\ \end{matrix} \right)$, ${{l}_{2}}\mathrm{:}x+3=1-y,z=0$

Relationships Between Two Lines

<p style="margin-bottom: 1.5rem">Determine whether the following pair of lines are parallel, intersecting or skew. Find the position vector of the intersection point (if any) and the angle between the lines.</p>
<p>${{l}_{1}}\mathrm{:}x=y-1=z-6$, ${{l}_{2}}\mathrm{:}\frac{x-2}{3}=\frac{z-3}{2};y=2$</p>

${{l}_{1}}\mathrm{:}x=y-1=z-6$, ${{l}_{2}}\mathrm{:}\frac{x-2}{3}=\frac{z-3}{2};y=2$

<p>Determine whether the following pair of lines are parallel, intersecting or skew. Find the position vector of the intersection point (if any) and the angle between the lines.</p>
<p>${{l}_{1}}\mathrm{:}z+2=\frac{y+1}{2}=x$,  ${{l}_{2}}\mathrm{:}x-5=-y,z=0$</p>

${{l}_{1}}\mathrm{:}z+2=\frac{y+1}{2}=x$,  ${{l}_{2}}\mathrm{:}x-5=-y,z=0$

<p style="margin-bottom: 1.5rem">Determine whether the following pairs of lines are parallel, intersecting or skew. Find the position vector of the intersection point (if any) and the angle between the lines.</p>
<p>${{l}_{1}}\mathrm{:r}=\left( \begin{matrix}3\\1\\-5\\\end{matrix} \right)+\lambda \left( \begin{matrix} -1\\-2\\3\\\end{matrix} \right)$ , ${{l}_{2}}\mathrm{:r}=\left( \begin{matrix}-1\\0\\9\\\end{matrix} \right)+\mu \left( \begin{matrix}3\\6\\-9 \\\end{matrix} \right)$</p>

Determine whether the following pairs of lines are parallel, intersecting or skew. Find the position vector of the intersection point (if any) and the angle between the lines.

${{l}_{1}}\mathrm{:r}=\left( \begin{matrix}3\\1\\-5\\\end{matrix} \right)+\lambda \left( \begin{matrix} -1\\-2\\3\\\end{matrix} \right)$ , ${{l}_{2}}\mathrm{:r}=\left( \begin{matrix}-1\\0\\9\\\end{matrix} \right)+\mu \left( \begin{matrix}3\\6\\-9 \\\end{matrix} \right)$

<p>Air traffic controllers need to know the exact locations of aircraft. They use coordinates $\left( x,y,z \right)$, with units in kilometres, to locate individual aircraft relative to the base of the control tower which is at position $\left( 0,0,0 \right)$. The ground is horizontal and is modelled as the $x-y$ plane.</p>
<p>An aircraft flies along a path with equation $\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   4  \\
   3  \\
   1  \\
\end{matrix}  \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
   \begin{matrix}
   1  \\
   -2  \\
   2  \\
\end{matrix}  \\
\end{matrix} \right)$. The air traffic controllers are situated in the control tower $0.1$ km above the base of the control tower.</p>
<ol class="abc-list"><li data-list-item-id="item-1761383870169-qjiehbtrh" data-list-item-path="item-1761383870169-qjiehbtrh"><div class="list-item-paragraph">Due to the atmospheric conditions at the time, the air traffic controllers can only see aircraft that come within $4$ km of their position.</div><ol class="i-list"><li data-list-item-id="subitem-1761383875591-k20q090e5" data-list-item-path="item-1761383870169-qjiehbtrh/subitem-1761383875591-k20q090e5"><div class="list-item-paragraph">Find the coordinates of this aircraft when it is at its closest point to the air traffic controllers.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1761383876364-f8e6h5sy8" data-list-item-path="item-1761383870169-qjiehbtrh/subitem-1761383876364-f8e6h5sy8"><div class="list-item-paragraph">Hence determine whether the air traffic controllers will see this aircraft.<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>A drone flies along a path with equation</p>
<p style="text-align: center">$\frac{2x-3}{3}=y+1=\frac{z-k}{-1}$,</p>
<p>where&nbsp;$k$ is a positive constant.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1761383928578-fbq6sbozw" data-list-item-path="item-1761383928578-fbq6sbozw"><div class="list-item-paragraph">Determine the possible values of ${k}$ if the paths of the aircraft and the drone do not intersect.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761383929261-h9uthakvm" data-list-item-path="item-1761383929261-h9uthakvm"><div class="list-item-paragraph">Find the acute angle between the paths of the aircraft and the drone.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383929876-wckta8sjs" data-list-item-path="item-1761383929876-wckta8sjs"><div class="list-item-paragraph">It is now given that&nbsp;$k=2$.</div><div class="list-item-paragraph">The aircraft and the drone are observed on radar to be closest to each other when they are at the points $\left( 4,3,1 \right)$ and $\left( 4.2,0.8,0.2 \right)$ on their respective paths.</div><ol class="i-list"><li data-list-item-id="subitem-1761383950538-7sfogdcu0" data-list-item-path="item-1761383929876-wckta8sjs/subitem-1761383950538-7sfogdcu0"><div class="list-item-paragraph">Find the distance between the aircraft and the drone at this instant.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1761383952496-ecnsymbfu" data-list-item-path="item-1761383929876-wckta8sjs/subitem-1761383952496-ecnsymbfu"><div class="list-item-paragraph">By considering the vector between the aircraft and the drone at this instant, determine whether the distance found in part <strong>(d)(i)</strong> is the shortest distance between the two paths.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Air traffic controllers need to know the exact locations of aircraft. They use coordinates $\left( x,y,z \right)$, with units in kilometres, to locate individual aircraft relative to the base of the control tower which is at position $\left( 0,0,0 \right)$. The ground is horizontal and is modelled as the $x-y$ plane.

An aircraft flies along a path with equation $\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   4  \\
   3  \\
   1  \\
\end{matrix}  \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
   \begin{matrix}
   1  \\
   -2  \\
   2  \\
\end{matrix}  \\
\end{matrix} \right)$. The air traffic controllers are situated in the control tower $0.1$ km above the base of the control tower.

Magnitude of a Vector

Perpendicular Vectors Using the Scalar Product

N2024 P1 Q12

<p>For positive real numbers $a$ and $b$, the lines $l_1$ and $l_2$ have equations given by</p>
<p style="text-align: center">${l_1:\ \mathbf r=\begin{pmatrix}1\\2\\5\end{pmatrix}+\lambda\begin{pmatrix}-1\\0\\3\end{pmatrix} \text{ for}\hspace{0.5em} \lambda\in\mathbb R,  \text{ and}\hspace{0.5em}    l_2:\ \mathbf r= \begin{pmatrix}a\\9\\9\end{pmatrix}+\mu\begin{pmatrix}1\\7\\b\end{pmatrix} \text{ for}\hspace{0.5em}  \mu\in\mathbb R.}$</p>
<p style="text-align: center"></p>
<p style="text-align: center"></p>
<p>It is given that the two lines intersect each other.</p>
<ol class="i-list"><li data-list-item-id="item-1761382441614-3rkn2np4o" data-list-item-path="item-1761382441614-3rkn2np4o"><div class="list-item-paragraph">Show that&nbsp;$3a=b+2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761382442764-e34tbqod0" data-list-item-path="item-1761382442764-e34tbqod0"><div class="list-item-paragraph">Find the value of $b$ if the angle between the two lines is $\cos^{-1}\!\left(\dfrac{11}{\sqrt{660}}\right)$ radians.<span class="inline-marks">[3]</span></div></li></ol>
<p>The equation of the plane ${{\Pi }_{1}}$ that contains both lines is now given by $\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   3  \\
   -1  \\
   1  \\
\end{matrix}  \\
\end{matrix} \right)=6$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1761382470544-vxm9r4psy" data-list-item-path="item-1761382470544-vxm9r4psy"><div class="list-item-paragraph">A second plane ${{\Pi }_{2}}$ meets ${{\Pi }_{1}}$ in the line ${{l}_{1}}$ and contains the point $\left( 2,1,3 \right)$. Find an equation for&nbsp;${{\Pi }_{2}}$ in scalar product 	form.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761382475007-2py335i75" data-list-item-path="item-1761382475007-2py335i75"><div class="list-item-paragraph">Hence find the angle between the planes ${{\Pi }_{1}}$ and ${{\Pi }_{2}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761382475704-r1vy72qw0" data-list-item-path="item-1761382475704-r1vy72qw0"><div class="list-item-paragraph">The plane ${{\Pi }_{3}}$ is parallel to the plane ${{\Pi }_{1}}$. Given that the distance between both planes is $\frac{7}{\sqrt{11}}$ units and that&nbsp;${{\Pi }_{3}}$ is closer to the origin than ${{\Pi }_{1}}$, find a cartesian equation for&nbsp;${{\Pi }_{3}}$.<span class="inline-marks">[3]</span></div></li></ol>

For positive real numbers $a$ and $b$, the lines $l_1$ and $l_2$ have equations given by

${l_1:\ \mathbf r=\begin{pmatrix}1\\2\\5\end{pmatrix}+\lambda\begin{pmatrix}-1\\0\\3\end{pmatrix} \text{ for } \lambda\in\mathbb R,  \text{ and }    l_2:\ \mathbf r= \begin{pmatrix}a\\9\\9\end{pmatrix}+\mu\begin{pmatrix}1\\7\\b\end{pmatrix} \text{ for }  \mu\in\mathbb R.}$

It is given that the two lines intersect each other.

The equation of the plane ${{\Pi }_{1}}$ that contains both lines is now given by $\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   3  \\
   -1  \\
   1  \\
\end{matrix}  \\
\end{matrix} \right)=6$.

Angle Between Two Planes

2024 ASRJC Promo Q10

<p>The line ${{l}_{2}}$ passes through $A\left( 2,-3,2 \right)$ and is parallel to the direction vector $\left( \begin{matrix}  29  \\-2  \\ 10  \\\end{matrix} \right)$.</p>
<p>${{l}_{2}}$ intersects ${{l}_{1}}:\mathbf{r}=\left( \begin{matrix} 1  \\  -5  \\2  \\\end{matrix} \right)+\lambda \left( \begin{matrix}14  \\-2  \\  5  \\\end{matrix} \right);\lambda \in \mathbb{R}$ at $C$. Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the coordinates of $C$,</div></li><li><div class="list-item-paragraph">the acute angle between ${{l}_{1}}$ and ${{l}_{2}}$,</div></li><li><div class="list-item-paragraph">the vector line equation of mirror image of ${{l}_{2}}$ in ${{l}_{1}}$.</div></li></ol>

The line ${{l}_{2}}$ passes through $A\left( 2,-3,2 \right)$ and is parallel to the direction vector $\left( \begin{matrix}  29  \\-2  \\ 10  \\\end{matrix} \right)$.

${{l}_{2}}$ intersects ${{l}_{1}}:\mathbf{r}=\left( \begin{matrix} 1  \\  -5  \\2  \\\end{matrix} \right)+\lambda \left( \begin{matrix}14  \\-2  \\  5  \\\end{matrix} \right);\lambda \in \mathbb{R}$ at $C$. Find

<p></p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start"><h3><strong>Relationships Between</strong></h3></th><th style="text-align: left; --cell-justify: flex-start"><h3>Essential Skills Checklist</h3></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A point and a line</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Shortest distance of the point to the line</p><p>☐ Foot of perpendicular</p><p>☐ Mirror image of a point in a line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A line segment and a line</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Length of projection of a line segment on a line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ parallel lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ $2$ parallel lines</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Intersecting Lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Point of intersection</p><p>☐ Angle between $2$ intersecting lines</p><p>☐ Mirror image of a line in another line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ skew Lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Angle between them</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A point and a plane</h3><h3></h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Deduce a random point on plane</p><p>☐ Shortest distance of point to plane</p><p>☐ Foot of perpendicular on the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>The origin and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Equations of planes which contains the origin</p><p>☐ $x-y$, $y-z$ and $x-z$ planes</p><p>☐ Shortest distance of origin to the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>Line segment and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Length of projection of line segment to the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>Line and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Point of intersection</p><p>☐ Angle between the line and plane</p><p>☐ Mirror Image of line in plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Intersecting Planes</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Equation of intersecting line without using GC</p><p>☐ Equation of intersecting line by using GC</p><p>☐ Angle between the planes</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Parallel Planes</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Distance between $2$ parallel planes</p><p>☐ Finding equations of planes a fixed distance away from another parallel plane</p></td></tr></tbody></table></div></div>

Finding Points on a Plane

Angle Between a Line and a Plane

Intersection of a Line and a Plane

Line of Intersection of Two Planes

<p>The line ${{l}_{2}}$ passes through $A\left( 3,1,-2 \right)$ and is parallel to the direction vector $\left( \begin{matrix}7  \\15  \\1  \end{matrix} \right)$.</p>
<p>${{l}_{2}}$ intersects ${{l}_{1}}:\mathbf{r}=\left( \begin{matrix}  -12  \\ -22  \\ 11  \end{matrix} \right)+\lambda \left( \begin{matrix}4  \\ 4  \\ -7  \end{matrix} \right);\lambda \in \mathbb{R}$ at $C$.</p>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="item-1761378751696-96ltfaxa1" data-list-item-path="item-1761378751696-96ltfaxa1"><div class="list-item-paragraph">the coordinates of $C$,</div></li><li data-list-item-id="item-1761378752332-9hcz2jlns" data-list-item-path="item-1761378752332-9hcz2jlns"><div class="list-item-paragraph">the acute angle between ${{l}_{1}}$ and ${{l}_{2}}$,</div></li><li data-list-item-id="item-1761378753075-0u8uf0lai" data-list-item-path="item-1761378753075-0u8uf0lai"><div class="list-item-paragraph">the vector line equation of mirror image of ${{l}_{2}}$ in ${{l}_{1}}$.</div></li></ol>

The line ${{l}_{2}}$ passes through $A\left( 3,1,-2 \right)$ and is parallel to the direction vector $\left( \begin{matrix}7  \\15  \\1  \\\end{matrix} \right)$.

${{l}_{2}}$ intersects ${{l}_{1}}:\mathbf{r}=\left( \begin{matrix}  -12  \\ -22  \\ 11  \\\end{matrix} \right)+\lambda \left( \begin{matrix}4  \\ 4  \\ -7  \\\end{matrix} \right);\lambda \in \mathbb{R}$ at $C$.

<p>In the diagram, $O$ is centre of the square base $ABCD$ of a right pyramid, vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $AD$, $OV$ respectively. The length of $AB$ is 4 units and the length of $OV$ is $2h$ units. $P$, $Q$, $M$ and $N$ are mid-points of $AB$, $BC$, $CV$ and $VA$ respectively. The point $O$ is taken as the origin for position vectors.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761370065226-1nk23hwyjjt.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNzAwNjUyMjYtMW5rMjNod3lqanQuanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwODYzMSwiZXhwIjoxNzkzNTA4OTMxfQ.hHRpVISP2dItUhdiRoUQ1jQNVPV5eDCfK34AitxGqoM" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the equation of the line $PM$ may be expressed as $\mathbf{r}=\left( \begin{matrix} 0 \\ -2 \\ 0 \\
\end{matrix} \right)+t\left( \begin{matrix} 1 \\ 3 \\ h \\
\end{matrix} \right)$, where $t$ is a parameter.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the equation for the line $QN$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the lines $PM$ and $QN$ intersect, and that the position vector $\overrightarrow{OX}$ of their point of intersection is $\left( \begin{matrix}
\frac{1}{2} \\
-\frac{1}{2} \\
\frac{1}{2}h \\
\end{matrix} \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $OX$ is perpendicular to $VB$, find the value of $h$ and calculate the acute angle between $PM$ and $QN$, giving your answer correct to the nearest $0.1{}^\circ $.</div><p style="text-align: right;">[4]</p></li></ol>

In the diagram, $O$ is centre of the square base $ABCD$ of a right pyramid, vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $AD$, $OV$ respectively. The length of $AB$ is 4 units and the length of $OV$ is $2h$ units. $P$, $Q$, $M$ and $N$ are mid-points of $AB$, $BC$, $CV$ and $VA$ respectively. The point $O$ is taken as the origin for position vectors.

Math Question Bank