A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

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

<p>Relative to the origin $O$, two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that&nbsp;$|\mathbf{a}|=2$, $|\mathbf{b}|=1$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{37}$.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-0" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-0"><div class="list-item-paragraph">By considering the scalar product $(3\mathbf{a}-2\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})$, show that&nbsp;$\mathbf{a}\cdot\mathbf{b}=\dfrac{1}{4}$, and give the geometrical meaning of $|\mathbf{a}\cdot\mathbf{b}|$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-1" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-1"><div class="list-item-paragraph">By finding the cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$, deduce the exact value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-2" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-2"><div class="list-item-paragraph">Interpret geometrically the value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-3" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-3"><div class="list-item-paragraph">Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$, a vector equation of the line that passes through $O$ and bisects the angle $AOB$.<span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that $|\mathbf{a}|=2$, $|\mathbf{b}|=1$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{37}$.

By considering the scalar product $(3\mathbf{a}-2\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})$, show that $\mathbf{a}\cdot\mathbf{b}=\dfrac{1}{4}$, and give the geometrical meaning of $|\mathbf{a}\cdot\mathbf{b}|$.

By finding the cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$, deduce the exact value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.

Interpret geometrically the value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.

Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$, a vector equation of the line that passes through $O$ and bisects the angle $AOB$.

Projection Lengths

Angles Between Vectors

Angle Bisectors Using Vectors

Equations of Lines

Areas of Triangles and Parallelograms Using Vectors

Scalar (Dot) Product

2025 SAJC Promo Q8

<ol class="abc-list"><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-0" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-0"><div class="list-item-paragraph">The non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are such that&nbsp;$3\mathbf{a}\times\mathbf{b}=2\mathbf{b}\times\mathbf{c}$, where&nbsp;$3\mathbf{a}\ne -2\mathbf{c}$. Find a linear relationship between $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1"><div class="list-item-paragraph"><p>The variable position vector $\mathbf{v}$ satisfies the equation</p></div><div class="list-item-paragraph"><p style="text-align: center;">$\mathbf{v}\times(-\mathbf{i}+2\mathbf{j}+\mathbf{k})=a\mathbf{i}+\mathbf{j}+\mathbf{k},$</p></div><div class="list-item-paragraph"><p>where $a$ is a non-zero constant.</p></div><ol class="i-list"><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1/h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-0"><div class="list-item-paragraph">Show that&nbsp;$a=3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1/h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-1"><div class="list-item-paragraph">Hence, by finding the set of position vectors $\mathbf{v}$, describe geometrically the set of points represented by the position vectors $\mathbf{v}$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

The non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are such that $3\mathbf{a}\times\mathbf{b}=2\mathbf{b}\times\mathbf{c}$, where $3\mathbf{a}\ne -2\mathbf{c}$. Find a linear relationship between $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.

Parallel Vectors

Perpendicular Vectors Using the Scalar Product

Geometrical Interpretation of Vector Equations

2025 RI Promo Q6

<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.

Finding Points on a Plane

Magnitude of a Vector

2025 NYJC Promo Q7

<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.

Unit Vectors

2025 EJC Promo Q3

<p>Two vectors $\mathbf{a}$ and $\mathbf{b}$ are such that&nbsp;$\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3$. It is given that&nbsp;$\mathbf{a}$ is a unit vector and $\mathbf{b} \cdot \mathbf{b} = 9$.</p>
<p>Show that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are perpendicular.<span class="inline-marks">[3]</span></p>

Two vectors $\mathbf{a}$ and $\mathbf{b}$ are such that $\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3$. It is given that $\mathbf{a}$ is a unit vector and $\mathbf{b} \cdot \mathbf{b} = 9$.

Show that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

N2024 P1 Q2

<p>Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and they lie on plane $\pi$ .</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;$\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}$ is a vector perpendicular to the plane $\pi$ .<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">Prove that the equation of plane $\pi$ can be written as</div><div class="list-item-paragraph" style="text-align: center">\[\mathbf{r}\bullet(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}) = \mathbf{a}\bullet(\mathbf{b}\times\mathbf{c}),\]</div><div class="list-item-paragraph">explaining clearly the reason for any result that you use in your proof.<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">Given that&nbsp;$\mathbf{a} = \mathbf{i}$, $\mathbf{b} = \mathbf{j}$ and $\mathbf{c} = \mathbf{k}$, show that the equation of the plane $\pi$ can be written as&nbsp;$\mathbf{r}\bullet\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 1$. Hence, find the cartesian equations of the planes which are at a distance of 5 units from plane $\pi$.<span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and they lie on plane $\pi$ .

2024 ACJC P2 Q2

<p>The position vectors of points $A$, $B$ and $C$, relative to the origin $O$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively, where</p>
<p style="text-align: center">\[\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k},\quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k},\quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find $\mathbf{a} \times \mathbf{b}$ and deduce the exact area of triangle $OAB$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Hence, find the volume of the tetrahedron $OABC$.</div><div class="list-item-paragraph">[The volume of a tetrahedron is $\frac{1}{3} \times \text{base area} \times \text{perpendicular height}$.]<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Find the perpendicular distance from $B$ to line $OA$.<span class="inline-marks">[2]</span></div></li></ol>

The position vectors of points $A$, $B$ and $C$, relative to the origin $O$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively, where

\[\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k},\quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k},\quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.\]

Three-Dimensional Geometry

Distance from a Point to a Line

2024 NYJC Promo Q5

<p>In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$ and $\overrightarrow{PR}$ respectively.</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/1779426366202-yu4qd1rzjpn.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0MjYzNjYyMDIteXU0cWQxcnpqcG4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDc0MCwiZXhwIjoxNzkzNTA1MDQwfQ.Pq60cw6YVF43naUiB72PRw7zd3x1ontHt1Zmu9N9qoM" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">State a geometrical interpretation of $|\mathbf{c} \times \mathbf{\hat{a}}|$ and hence show that the area of the triangle $PQR$ can be written as&nbsp;$\frac{1}{2}|\mathbf{c} \times \mathbf{{a}}|$ units$^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With clear explanations, show that&nbsp;$|\mathbf{c} \times \mathbf{a}| = |\mathbf{b} \times \mathbf{a}|$ and deduce that&nbsp;$\frac{|\mathbf{c}|}{\sin \angle PQR} = \frac{|\mathbf{b}|}{\sin \angle QPR}$ (Sine Rule).<span class="inline-marks">[2]</span></div></li></ol>

In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$ and $\overrightarrow{PR}$ respectively.

2024 ASRJC Promo Q5

<p></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The non-zero vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that&nbsp;$\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a}$. Given that&nbsp;$\mathbf{b} \neq -\mathbf{c}$, find a linear relationship between $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The variable position vector $\mathbf{v}$ satisfies the equation $\mathbf{v} \times (\mathbf{i} - 3\mathbf{k}) = 2\mathbf{j}$.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the set of position vectors $\mathbf{v}$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe the set of points represented by position vectors $\mathbf{v}$ geometrically.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

The non-zero vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a}$. Given that $\mathbf{b} \neq -\mathbf{c}$, find a linear relationship between $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.

2025 Specimen Paper P1 Q6

<p>For any vectors $\mathbf{m}$ and $\mathbf{n}$, explain why $\mathbf{m} \cdot (\mathbf{m} \times \mathbf{n}) = 0$.<span class="inline-marks">[1]</span></p>
<p>With respect to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. $O$, $A$, $B$ and $C$ are non-coplanar. The point $M$ is the mid-point of $AC$ and $p$ denotes the plane $OAB$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The point $R$ is such that&nbsp;$MR$ is perpendicular to $p$. Show that&nbsp;$R$ lies on a line $\ell$ with equation $\mathbf{r} = k(\mathbf{a}+\mathbf{c}) + \lambda(\mathbf{a} \times \mathbf{b})$, $\lambda \in \mathbb{R}$, where&nbsp;$k$ is a constant to be determined.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are unit vectors perpendicular to each other, $\mathbf{a} \cdot \mathbf{c} = -2$ and $\mathbf{b} \cdot \mathbf{c} = 4$, find in terms of $\mathbf{a}$ and $\mathbf{b}$, the position vector of the point of intersection between $\ell$ and $p$.</span><span class="inline-marks">[3]</span></div></li></ol>

For any vectors $\mathbf{m}$ and $\mathbf{n}$, explain why $\mathbf{m} \cdot (\mathbf{m} \times \mathbf{n}) = 0$.

With respect to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. $O$, $A$, $B$ and $C$ are non-coplanar. The point $M$ is the mid-point of $AC$ and $p$ denotes the plane $OAB$.

The point $R$ is such that $MR$ is perpendicular to $p$. Show that $R$ lies on a line $\ell$ with equation $\mathbf{r} = k(\mathbf{a}+\mathbf{c}) + \lambda(\mathbf{a} \times \mathbf{b})$, $\lambda \in \mathbb{R}$, where $k$ is a constant to be determined.

Given that $\mathbf{a}$ and $\mathbf{b}$ are unit vectors perpendicular to each other, $\mathbf{a} \cdot \mathbf{c} = -2$ and $\mathbf{b} \cdot \mathbf{c} = 4$, find in terms of $\mathbf{a}$ and $\mathbf{b}$, the position vector of the point of intersection between $\ell$ and $p$.

Perpendicular Projection and Reflection in a Plane

Ratio Theorem and Section Formula

2025 RVHS P1 Q6

<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 $|\mathbf{a} \times \mathbf{b}| = \frac{1}{4}(|\mathbf{a}+2\mathbf{b}|)(|\mathbf{a}-2\mathbf{b}|)$.

Vectors in a Parallelogram

2025 RI P1 Q3

<p>With respect to an origin $O$, points $P$ and $Q$ have variable position vectors $\mathbf{p}$ and $\mathbf{q}$ respectively, given by $\mathbf{p}=(\cos t)\mathbf{i}+(\sin t)\mathbf{j}-\mathbf{k}$, $\mathbf{q}=(\cos 2t)\mathbf{i}-(\sin 2t)\mathbf{j}+\alpha \mathbf{k}$, where&nbsp;$\alpha$ is a real parameter and $0&lt;t&lt;\pi$.</p>
<ol class="i-list"><li data-list-item-id="item-i-4" data-list-item-path="item-i-4"><div class="list-item-paragraph">By considering the cross product or otherwise, find the exact values of $t$ and $\alpha$ if&nbsp;$O$, $P$ and $Q$ are collinear.<span class="inline-marks">[5]</span></div></li></ol>
<p>It is given that&nbsp;$\alpha=0.5$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii-4" data-list-item-path="item-ii-4"><div class="list-item-paragraph">Show that&nbsp;$\mathbf{p}\cdot \mathbf{q}=\cos 3t-0.5$. Find the greatest length of projection of $\mathbf{q}$ onto $\mathbf{p}$.<span class="inline-marks">[4]</span></div></li></ol>

With respect to an origin $O$, points $P$ and $Q$ have variable position vectors $\mathbf{p}$ and $\mathbf{q}$ respectively, given by $\mathbf{p}=(\cos t)\mathbf{i}+(\sin t)\mathbf{j}-\mathbf{k}$, $\mathbf{q}=(\cos 2t)\mathbf{i}-(\sin 2t)\mathbf{j}+\alpha \mathbf{k}$, where $\alpha$ is a real parameter and $0<t<\pi$.

By considering the cross product or otherwise, find the exact values of $t$ and $\alpha$ if $O$, $P$ and $Q$ are collinear.

Show that $\mathbf{p}\cdot \mathbf{q}=\cos 3t-0.5$. Find the greatest length of projection of $\mathbf{q}$ onto $\mathbf{p}$.

Collinearity Using Vectors

2025 NJC P2 Q3

<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 $O$, 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>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>Find the Parametric and Cartesian equations of line which passes through the origin and perpendicular to both&nbsp;$(-\mathbf{k})$ and $(-2\mathbf{i}-\mathbf{j}+8\mathbf{k})$.</p>

Find the Parametric and Cartesian equations of line which passes through the origin and perpendicular to both $(-\mathbf{k})$ and $(-2\mathbf{i}-\mathbf{j}+8\mathbf{k})$.

<p>Find the Parametric and Cartesian equations of line which passes through the points $A(-2\mathbf{i} + \mathbf{j} + 3\mathbf{k})$ and perpendicular to both&nbsp;$(\mathbf{i} - 3\mathbf{k})$ and $(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})$.</p>

Find the Parametric and Cartesian equations of line which passes through the points $A(-2\mathbf{i} + \mathbf{j} + 3\mathbf{k})$ and perpendicular to both $(\mathbf{i} - 3\mathbf{k})$ and $(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})$.

<p>Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=3\mathbf{i}+4\mathbf{j}-\mathbf{k}$, find:</p>
<ol class="i-list"><li><div class="list-item-paragraph">a unit vector perpendicular to both&nbsp;$\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">the exact value of $\sin\theta$, where&nbsp;$\theta$ 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 $\mathbf{b}=3\mathbf{i}+4\mathbf{j}-\mathbf{k}$, find:

<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&nbsp;$\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">the exact value of $\sin\theta$, where&nbsp;$\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.</div></li></ol>

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