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PSLE

PSLE Foundation Mathematics

<p>Let&nbsp;$\theta$ be the angle between the unit vectors $ \mathbf{a}$ and $ \mathbf{b}$, where&nbsp;$0 &lt; \theta &lt; \pi$. Express $\vert{}\mathbf{a} - \mathbf{b}\vert{}$ in terms of $\sin \frac{1}{2}\theta$</p>

Let $\theta$ be the angle between the unit vectors $ \mathbf{a}$ and $ \mathbf{b}$, where $0 < \theta < \pi$. Express $\vert{}\mathbf{a} - \mathbf{b}\vert{}$ in terms of $\sin \frac{1}{2}\theta$

Magnitude of a Vector

Scalar (Dot) Product

Vectors

<ol class="abc-list"><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-0" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-0"><div class="list-item-paragraph">Given that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that&nbsp;$\mathbf{a}=(\mathbf{a}\cdot\mathbf{b})\mathbf{b}$. State the relationship between $\mathbf{a}$ and $\mathbf{b}$. Find $|\mathbf{b}|$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1"><div class="list-item-paragraph">With reference to the origin $O$, the points $M$ and $N$ are such that&nbsp;$\overrightarrow{OM}=\alpha\mathbf{i}+4\mathbf{j}+\beta\mathbf{k}$ and $\overrightarrow{ON}=4\mathbf{i}+\alpha\mathbf{j}+\beta\mathbf{k}$, where&nbsp;$\alpha$ and $\beta$ are constants.</div><ol class="i-list"><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1/h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-0"><div class="list-item-paragraph">Given that&nbsp;$M$ lies in the $xy$-plane and $|\overrightarrow{OM}|=5$, find the values of $\alpha$ and $\beta$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1/h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-1"><div class="list-item-paragraph"><p>Given <strong>instead</strong> that&nbsp;$\overrightarrow{OM}$ is perpendicular to $\overrightarrow{ON}$, find an expression for&nbsp;$|\overrightarrow{OM}\times\overrightarrow{ON}|$ in terms of $\alpha$, giving your answer in fully factorised form.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>

Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that $\mathbf{a}=(\mathbf{a}\cdot\mathbf{b})\mathbf{b}$. State the relationship between $\mathbf{a}$ and $\mathbf{b}$. Find $|\mathbf{b}|$.

With reference to the origin $O$, the points $M$ and $N$ are such that $\overrightarrow{OM}=\alpha\mathbf{i}+4\mathbf{j}+\beta\mathbf{k}$ and $\overrightarrow{ON}=4\mathbf{i}+\alpha\mathbf{j}+\beta\mathbf{k}$, where $\alpha$ and $\beta$ are constants.

Parallel Vectors

Finding Points on a Plane

Vector (Cross) Product

Perpendicular Vectors Using the Scalar Product

2025 NYJC Promo Q7

<p>Find a unit vector $\mathbf{d}$ such that&nbsp;$\left|\mathbf{c}-\mathbf{d}\right|=\left|\mathbf{c}\right|+\left|\mathbf{d}\right|$ if&nbsp;$\mathbf{c}=\begin{pmatrix}-2 \\ 2 \\ 1\end{pmatrix}$.<span class="inline-marks">[2]</span></p>

Find a unit vector $\mathbf{d}$ such that $\left|\mathbf{c}-\mathbf{d}\right|=\left|\mathbf{c}\right|+\left|\mathbf{d}\right|$ if $\mathbf{c}=\begin{pmatrix}-2 \\ 2 \\ 1\end{pmatrix}$.

Unit Vectors

<p>The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where&nbsp;$c$ is a constant, and $s$ and $t$ are parameters. The point $A(1,3,-2)$ lies in both planes.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$c=-1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Show that the vector equation of the line of intersection of $\pi_1$ and $\pi_2$, line $l$, is given by $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})$, where&nbsp;$\alpha$ is a parameter.<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 position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2,-3,7)$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$.</div><div class="list-item-paragraph">Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.<span class="inline-marks">[3]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where $c$ is a constant, and $s$ and $t$ are parameters. The point $A(1,3,-2)$ lies in both planes.

Relationships Between a Point and a Plane

Distance from a Point to a Plane

Equations of Planes

Line of Intersection of Two Planes

2025 ACJC P1 Q9

<p>The line $l$ has vector equation</p>
<p style="text-align: center">\[l : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} , \quad \lambda \in \mathbb{R}.\]</p>
<ol class="abc-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find the position vector of the point $N$, the foot of perpendicular from the point $A(6, 4, 4)$ to the line $l$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the position vector of the point $A'$, the reflection of $A$ in the line $l$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the coordinates of the points on the line $l$ that are $3\sqrt{3}$ units away from point $A$.<span class="inline-marks">[2]</span></div></li></ol>

\[l : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} , \quad \lambda \in \mathbb{R}.\]

Perpendicular Projection and Reflection in a Line

<p>$\mathbf{a}$ and $\mathbf{b}$ are vectors such that&nbsp;$|\mathbf{a}| = \sqrt{2}$ and $|\mathbf{b}| = \sqrt{6}$. The angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{5\pi}{6}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find the exact length of projection of $\mathbf{a}$ onto $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">By considering $(3\mathbf{a} + 2\mathbf{b}) \cdot (3\mathbf{a} + 2\mathbf{b})$, find the exact value of $|3\mathbf{a} + 2\mathbf{b}|$.<span class="inline-marks">[4]</span></div></li></ol>

$\mathbf{a}$ and $\mathbf{b}$ are vectors such that $|\mathbf{a}| = \sqrt{2}$ and $|\mathbf{b}| = \sqrt{6}$. The angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{5\pi}{6}$.

Projection Lengths

2024 EJC Promo Q5

<p>Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ such that</p>
<p style="text-align: center">\[\mathbf{a} = \mathbf{i} + \mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{b} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} .\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the size of angle $OAB$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The point $C$ has position vector $\mathbf{c}$ given by $\mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b}$, where&nbsp;$\lambda$ and $\mu$ are positive constants.<br>Given that the area of triangle $OAC$ is twice that of triangle $OBC$,</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">find $\mu$ in terms of $\lambda$,</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">hence, if&nbsp;$OC = \sqrt{118}$, find the position vector $\mathbf{c}$.</span><span class="inline-marks">[4]</span></div></li></ol>

Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ such that

\[\mathbf{a} = \mathbf{i} + \mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{b} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} .\]

The point $C$ has position vector $\mathbf{c}$ given by $\mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b}$, where $\lambda$ and $\mu$ are positive constants.
Given that the area of triangle $OAC$ is twice that of triangle $OBC$,

hence, if $OC = \sqrt{118}$, find the position vector $\mathbf{c}$.

Angles Between Vectors

Areas of Triangles and Parallelograms Using Vectors

2021 JPJC P1 Q2

<p>With respect to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are not parallel.</p>
<ol class="i-list"><li><div class="list-item-paragraph">It is given that&nbsp;$B$ lies on the line segment $AC$, such that&nbsp;$\overrightarrow{BC}=2\mathbf{b}-k\mathbf{a}$. State the value of the constant $k$. Hence, 	find $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$\overrightarrow{ON}$ is the projection vectors of $\overrightarrow{OB}$ onto $\overrightarrow{OA}$, where&nbsp;$\overrightarrow{ON}=\frac{3}{4}\mathbf{a}$ and $\mathbf{a}\cdot \mathbf{b}=4$. Show that the length of $OA$ is $\frac{4\sqrt{3}}{3}$.	<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given also that&nbsp;$\mathbf{b}=\left( \begin{matrix}
   2  \\
   -2  \\
   2  \\
\end{matrix} \right)$, find the angle between $\overrightarrow{OA}$ and  $\overrightarrow{OB}$.	<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$OADC$ is a parallelogram and $E$ lies on line segment $AC$ such that&nbsp;$AE:AC=2:3$, find $\overrightarrow{DE}$.<span class="inline-marks">[3]</span></div></li></ol>

With respect to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are not parallel.

Vector Projections

Collinearity Using Vectors

Vectors in a Parallelogram

Ratio Theorem and Section Formula

<p>Given&nbsp;$\left| \mathbf{a} \right|=7$ and $\left| \mathbf{b} \right|=4$, $\mathbf{a}\cdot \mathbf{b}=8$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the angle between vectors $\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">Find the values of $(\mathbf{a}+\mathbf{b})\cdot (\mathbf{a}+\mathbf{b})$ and $(\mathbf{a}-\mathbf{b})\cdot (\mathbf{a}-\mathbf{b})$.</div></li><li><div class="list-item-paragraph">Hence, find the values of $\left| \mathbf{a}+\mathbf{b} \right|$ and $\left| \mathbf{a}-\mathbf{b} \right|$.</div></li></ol>

Given $\left| \mathbf{a} \right|=7$ and $\left| \mathbf{b} \right|=4$, $\mathbf{a}\cdot \mathbf{b}=8$.

<p>Relative to the origin $O$, the points $A$ and $C$ have position vectors $\mathbf{a}$ and $2\mathbf{b}$ such that</p>
<p style="text-align: center">\[\mathbf{a}=5p\mathbf{i}-2p\mathbf{j}+4p\mathbf{k} \text{ and}\hspace{0.5em} \mathbf{b}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},\]</p>
<p>where&nbsp;$p$ is a positive constant. It is given that&nbsp;$|\mathbf{a}|=2|\mathbf{b}|$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact value of $p$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Evaluate $(\mathbf{a}+2\mathbf{b}) \cdot (\mathbf{a}-2\mathbf{b})$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Evaluate $\mathbf{a} \times \mathbf{b}$ and hence find the area of triangle $OAC$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Use a geometrical reason to explain why $|\mathbf{a} \times \mathbf{b}| = \frac{1}{4}(|\mathbf{a}+2\mathbf{b}|)(|\mathbf{a}-2\mathbf{b}|)$.</span><span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, the points $A$ and $C$ have position vectors $\mathbf{a}$ and $2\mathbf{b}$ such that

\[\mathbf{a}=5p\mathbf{i}-2p\mathbf{j}+4p\mathbf{k} \text{ and } \mathbf{b}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},\]

where $p$ is a positive constant. It is given that $|\mathbf{a}|=2|\mathbf{b}|$.

Evaluate $(\mathbf{a}+2\mathbf{b}) \cdot (\mathbf{a}-2\mathbf{b})$.

Evaluate $\mathbf{a} \times \mathbf{b}$ and hence find the area of triangle $OAC$.

Use a geometrical reason to explain why $|\mathbf{a} \times \mathbf{b}| = \frac{1}{4}(|\mathbf{a}+2\mathbf{b}|)(|\mathbf{a}-2\mathbf{b}|)$.

2025 RI P1 Q3

<p>The points $A$ and $B$ have coordinates $(15, 3, 0)$ and $(5, 9, 5)$ respectively. Two lines $l_1$ and $l_2$, which are perpendicular to each other, have the following equations.</p>
<p style="margin-left: 7rem">$l_1 : \mathbf{r} = (15 - \lambda)\mathbf{i} + (3 + 2\lambda)\mathbf{j} + 4\lambda\mathbf{k},$<br>$l_2 : \mathbf{r} = (5 + 8\mu)\mathbf{i} + (9 - 2\mu)\mathbf{j} + (5 + m\mu)\mathbf{k},$</p>
<p>where&nbsp;$\lambda$ and $\mu$ are parameters and $m$ is a constant.</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 value of $m$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Given that&nbsp;$l_1$ and $l_2$ intersect, find the coordinates of the point of intersection $E$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find a cartesian equation of the plane $\Pi$ which contains the points $A$, $B$ and $E$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">The point $D$ has coordinates $(-1, -3, 2)$. Find the position vector of the point $F$, the foot of perpendicular of $D$ to $\Pi$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Find the exact area of the circle that passes through $A$, $D$ and $F$.<span class="inline-marks">[2]</span></div></li></ol>

The points $A$ and $B$ have coordinates $(15, 3, 0)$ and $(5, 9, 5)$ respectively. Two lines $l_1$ and $l_2$, which are perpendicular to each other, have the following equations.

$l_1 : \mathbf{r} = (15 - \lambda)\mathbf{i} + (3 + 2\lambda)\mathbf{j} + 4\lambda\mathbf{k},$
$l_2 : \mathbf{r} = (5 + 8\mu)\mathbf{i} + (9 - 2\mu)\mathbf{j} + (5 + m\mu)\mathbf{k},$

where $\lambda$ and $\mu$ are parameters and $m$ is a constant.

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

Find a cartesian equation of the plane $\Pi$ which contains the points $A$, $B$ and $E$.

The point $D$ has coordinates $(-1, -3, 2)$. Find the position vector of the point $F$, the foot of perpendicular of $D$ to $\Pi$.

Find the exact area of the circle that passes through $A$, $D$ and $F$.

Perpendicular Projection and Reflection in a Plane

Intersection of Two Lines

2025 HCI P1 Q11

<p>Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. It is given that&nbsp;$\mathbf{a}$ is a unit vector and $\mathbf{b}$ is perpendicular to $\mathbf{b} - \mathbf{a}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$0 &lt; |\mathbf{b}| &lt; 1$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is given further that&nbsp;$|\mathbf{b}| = \frac{1}{2}$ and $C$ lies on $AB$ such that&nbsp;$AC : CB = 2 : 1$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the value of $|\mathbf{a} \cdot \mathbf{c}|$ and state the geometrical interpretation of $|\mathbf{a} \cdot \mathbf{c}|$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \times \mathbf{c}|}$.</span><span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. It is given that $\mathbf{a}$ is a unit vector and $\mathbf{b}$ is perpendicular to $\mathbf{b} - \mathbf{a}$.

It is given further that $|\mathbf{b}| = \frac{1}{2}$ and $C$ lies on $AB$ such that $AC : CB = 2 : 1$.

Find the value of $|\mathbf{a} \cdot \mathbf{c}|$ and state the geometrical interpretation of $|\mathbf{a} \cdot \mathbf{c}|$.

Find the value of $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \times \mathbf{c}|}$.

2025 CJC P1 Q6

<p>The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where&nbsp;$c$ is a constant, and $s$ and $t$ are parameters. The point $A(1, 3, -2)$ lies in both planes.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$c=-1$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the vector equation of the line of intersection of $\pi_1$ and $\pi_2$, line $l$, is given by $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})$, where&nbsp;$\alpha$ is a parameter.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2, -3, 7)$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$. Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.</span><span class="inline-marks">[3]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where $c$ is a constant, and $s$ and $t$ are parameters. The point $A(1, 3, -2)$ lies in both planes.

Show that the vector equation of the line of intersection of $\pi_1$ and $\pi_2$, line $l$, is given by $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})$, where $\alpha$ is a parameter.

Find the position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2, -3, 7)$.

Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$. Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.

<p>In the diagram, $OABC$ is a parallelogram. $D$ is the mid-point of $BC$. It is given that&nbsp;$\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$. $E$ is the point on $OC$ such that&nbsp;$5OE = 2OC$. $F$ is the point on $OD$ such that&nbsp;$\overrightarrow{OF} = \frac{1}{2}\overrightarrow{FD}$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Express, as simply as possible, in terms of $\mathbf{a}$ and/or $\mathbf{c}$,</div><ol class="i-list"><li><div class="list-item-paragraph">$\overrightarrow{EC}$,</div></li><li><div class="list-item-paragraph">$\overrightarrow{OD}$,</div></li><li><div class="list-item-paragraph">$\overrightarrow{OF}$.</div></li></ol></li><li><div class="list-item-paragraph">Show that&nbsp;$\overrightarrow{AE} = 6\overrightarrow{FE}$.</div></li><li><div class="list-item-paragraph">Given that the coordinates of $F$ are $(4,2)$ and $\overrightarrow{AE} = \begin{pmatrix} 3 \\ -12 \end{pmatrix}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Calculate $|\overrightarrow{AE}|$.</div></li><li><div class="list-item-paragraph">Find the coordinates of $E$.</div></li><li><div class="list-item-paragraph">Find the value of $t$, given that the vector $\begin{pmatrix} 4.5 \\ t \end{pmatrix}$ is parallel to $\overrightarrow{AE}$.</div></li></ol></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1768037372726-zd47zjcpps.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMzczNzI3MjYtemQ0N3pqY3Bwcy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0Nzk4LCJleHAiOjE3OTM1MDUwOTh9.2JNpPl3J597MBheYhlROuf9h6AiXK_mRLd38n4iZKV8" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, $OABC$ is a parallelogram. $D$ is the mid-point of $BC$. It is given that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$. $E$ is the point on $OC$ such that $5OE = 2OC$. $F$ is the point on $OD$ such that $\overrightarrow{OF} = \frac{1}{2}\overrightarrow{FD}$.

Position Vectors

Vector Properties in Geometric Shapes: The Parallelogram

<p>The coordinates of point $A$ and point $B$ are $(5,3)$ and $(2.5,-1)$ respectively.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$\overrightarrow{OC} = \overrightarrow{OA} - 3\overrightarrow{OB}$, find the column vector $\overrightarrow{OC}$.</span></div></li><li><div class="list-item-paragraph">$\overrightarrow{ST} = \begin{pmatrix} a \\ b \end{pmatrix}$, $|\overrightarrow{ST}| = 13$ units and $\overrightarrow{ST}$ is parallel to $\overrightarrow{OC}$. Find the value of $a$ and of $b$ if&nbsp;$a &gt; 0$.</div></li></ol>

The coordinates of point $A$ and point $B$ are $(5,3)$ and $(2.5,-1)$ respectively.

Given that $\overrightarrow{OC} = \overrightarrow{OA} - 3\overrightarrow{OB}$, find the column vector $\overrightarrow{OC}$.

Vector Algebra

Resultant Vector Calculations in Cartesian Form

<p>It is given that&nbsp;$A$ is the point $(3,4)$ and $B$ is the point $(-5,1)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $|\overrightarrow{AB}|$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\overrightarrow{CD} = \frac{1}{2}\overrightarrow{AB}$ and $D$ is the point $(7,2)$, find the coordinates of the point $C$.</span></div></li><li><div class="list-item-paragraph">If&nbsp;$\overrightarrow{XY} = \begin{pmatrix} -2 \\ k \end{pmatrix}$ and $\overrightarrow{XY}$ is parallel to $\overrightarrow{AB}$, find the value of $k$.</div></li></ol>

It is given that $A$ is the point $(3,4)$ and $B$ is the point $(-5,1)$.

If $\overrightarrow{CD} = \frac{1}{2}\overrightarrow{AB}$ and $D$ is the point $(7,2)$, find the coordinates of the point $C$.

Algebraic Manipulation of Planar Vectors

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given the coordinates $A(-1,3)$ and $B(0,6)$, express $\overrightarrow{AB}$ as a column vector.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, evaluate $\left| \overrightarrow{AB} \right|$.</span></div></li></ol>

Given the coordinates $A(-1,3)$ and $B(0,6)$, express $\overrightarrow{AB}$ as a column vector.

Hence, evaluate $\left| \overrightarrow{AB} \right|$.

Vector Subtraction

Operations on Vectors: Addition and Subtraction

<p>Given that&nbsp;$\mathbf{p} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}$, $\mathbf{q} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} m \\ -2 \end{pmatrix}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $|\mathbf{q}|$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Express $2\mathbf{p} - \mathbf{q}$ as a column vector.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$\mathbf{p}$ is parallel to $\mathbf{r}$, find $m$.</span></div></li></ol>

Given that $\mathbf{p} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}$, $\mathbf{q} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix} m \\ -2 \end{pmatrix}$.

Express $2\mathbf{p} - \mathbf{q}$ as a column vector.

Given that $\mathbf{p}$ is parallel to $\mathbf{r}$, find $m$.

Calculating Unknown Variables in Parallel Vector Systems

<p>Given that&nbsp;$\mathbf{a} = \begin{pmatrix} -12 \\ 5 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 6 \\ -k \end{pmatrix}$,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find $|\mathbf{a}|$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$\mathbf{a}$ is parallel to $\mathbf{b}$, find the value of $k$.</span></div></li></ol>

Given that $\mathbf{a} = \begin{pmatrix} -12 \\ 5 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 6 \\ -k \end{pmatrix}$,

Given that $\mathbf{a}$ is parallel to $\mathbf{b}$, find the value of $k$.

Determining Parallelism in 2D Column Vectors

<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/1768028200059-ds6p82abwbu.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMjgyMDAwNTktZHM2cDgyYWJ3YnUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDc5OCwiZXhwIjoxNzkzNTA1MDk4fQ.rCeVRtzV9uuBbXU910JLhfRTsxe3WCLwBsoM2J5hzmQ" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Square grid with x and y axes" class="aligncenter custom-width" /></figure></div>
<p>Given that column vectors $\mathbf{a} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 10 \\ -4 \end{pmatrix}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Draw, on the $1$ unit by $1$ unit square grid provided, the relationship between $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the column vector of $\mathbf{a} - \mathbf{b}$ numerically.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of $|\mathbf{a}|$, $|\mathbf{b}|$ and $|\mathbf{a} - \mathbf{b}|$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Is $|\mathbf{a} - \mathbf{b}| = |\mathbf{a}| - |\mathbf{b}|$? Explain your answer.</span></div></li></ol>

Given that column vectors $\mathbf{a} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 10 \\ -4 \end{pmatrix}$.

Draw, on the $1$ unit by $1$ unit square grid provided, the relationship between $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$.

Find the column vector of $\mathbf{a} - \mathbf{b}$ numerically.

Find the value of $|\mathbf{a}|$, $|\mathbf{b}|$ and $|\mathbf{a} - \mathbf{b}|$.

Is $|\mathbf{a} - \mathbf{b}| = |\mathbf{a}| - |\mathbf{b}|$? Explain your answer.

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