Vectors in two dimensions

The points $A$ and $B$ are shown on the grid, and $\vec{BC} = \begin{pmatrix}-4 \\ 0\end{pmatrix}$.

Feb/March 2017

The sketch depicts the regular hexagon $ABCDEF$. Also, $\overrightarrow{CD}=p$ and $\overrightarrow{CB}=q$.

Feb/March 2018

Find the position vector of $B$, in terms of $x$ and $y$, in the simplest form.

Feb/March 2019

Point $A$ is located at $(6, 4)$ and point $B$ is located at $(2, 7)$.

Feb/March 2020

Vector v is $\begin{pmatrix}-1\\3\end{pmatrix}$ vector y is $\begin{pmatrix}2\\5\end{pmatrix}$

Feb/March 2023

The coordinate diagram displays points $A$ and $B$ on axes named $x$ and $y$.

Feb/March 2024

The vectors are $p = \begin{pmatrix}8\\-5\end{pmatrix}$ and $q = \begin{pmatrix}-4\\5\end{pmatrix}$.

Feb/March 2024

Point $A$ is located at $(-4, 1)$, and the vector $\vec{BA}$ is given by $\begin{pmatrix}-5\\-12\end{pmatrix}$.

Feb/March 2025

PQRS is a quadrilateral, and M lies at the midpoint of PS. $\overrightarrow{PQ} = \mathbf{a}$, $\overrightarrow{QR} = \mathbf{b}$ and $\overrightarrow{SQ} = \mathbf{a} - 2\mathbf{b}$. Diagram labels: points P, Q, R, S and M, together with vectors marked $a$ and $b$. The diagram is NOT TO SCALE.

May/June 2015

Vector algebra and matrix transformations.

May/June 2015

In the diagram shown, $O$ is the origin, $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OC} = \mathbf{c}$ and $\overrightarrow{AB} = \mathbf{b}$. $P$ lies on the line $AB$ such that $AP : PB = 2 : 1$. $Q$ is the midpoint of $BC$. The diagram is marked NOT TO SCALE.

May/June 2016

Find the value of $5\vec{GH}$.

May/June 2017

Find the value of $|\vec{OB}|$.

May/June 2018

OABC is a parallelogram with O at the origin. $CK = 2KB$ and $AL = LB$. $M$ is the midpoint of $KL$. $\overrightarrow{OA} = \mathbf{p}$ and $\overrightarrow{OC} = \mathbf{q}$. The diagram has been drawn NOT TO SCALE.

May/June 2019

$ABCD$ forms a parallelogram. $N$ lies on $BD$ so that $BN : ND = 4 : 1$. $AB = s$ and $AD = t$. The sketch is not drawn to scale.

May/June 2019

Calculate $\begin{pmatrix}6\\-5\end{pmatrix}+\begin{pmatrix}8\\-1\end{pmatrix}$.

May/June 2021

The vectors are $\mathbf{a}=\begin{pmatrix}-3\\8\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}2\\-5\end{pmatrix}$.

May/June 2021

Find the coordinates of point $M$.

May/June 2021

The diagram depicts triangle $OAB$ together with parallelogram $OALK$. The position vector of $A$ is $\mathbf{a}$ and the position vector of $B$ is $\mathbf{b}$. Point $K$ lies on $AB$ so that $AK : KB = 1 : 2$. Sketch NOT TO SCALE.

May/June 2022

$p = \begin{pmatrix}2\\8\end{pmatrix}$ and $q = \begin{pmatrix}-1\\4\end{pmatrix}$.

May/June 2022

$F$ is located at $(1, -4)$, while $\vec{FG} = \begin{pmatrix}8\\-3\end{pmatrix}$ and $\vec{GH} = \begin{pmatrix}-12\\35\end{pmatrix}$.

May/June 2023

$m = \begin{pmatrix}11 \\ 5\end{pmatrix}$ and $n = \begin{pmatrix}8 \\ -3\end{pmatrix}$.

May/June 2025

$A$ is located at $(3,-1)$. The vector $verrightarrow{AB} = \begin{pmatrix}2\\-4\end{pmatrix}$.

May/June 2025

Calculate $5\mathbf{v}$.

May/June 2025

Here, $O$ denotes the origin, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Point $C$ lies on the straight line $AB$ with $AC : CB = 1 : 2$. The diagram is NOT TO SCALE.

Oct/Nov 2015

The vectors are specified as $\vec{BC}=\begin{pmatrix}2\\3\end{pmatrix}$ and $\vec{BA}=\begin{pmatrix}-5\\6\end{pmatrix}$.

Oct/Nov 2016

Here, $m = \begin{pmatrix}3 \\ 2\end{pmatrix}$ and $n = \begin{pmatrix}-2 \\ 3\end{pmatrix}$.

Oct/Nov 2016

O denotes the origin, and K lies on AB such that AK : KB = 2 : 1. $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$.

Oct/Nov 2017

The vectors are $a = \begin{pmatrix}-3\\2\end{pmatrix}$, $b = \begin{pmatrix}5\\4\end{pmatrix}$ and $c = \begin{pmatrix}14\\9\end{pmatrix}$.

Oct/Nov 2018

Make $p$ the subject in $5p + 7 = m$.

Oct/Nov 2019

The diagram indicates that $OAB$ is a triangle, while $ABC$ and $PQC$ are straight lines. $P$ is the midpoint of $OA$, $Q$ is the midpoint of $PC$, and $OQ : QB = 3 : 1$. Also, $\vec{OA} = 4a$ and $\vec{OB} = 8b$. NOT DRAWN TO SCALE.

Oct/Nov 2019

The vectors are supplied as $\overrightarrow{AB} = \begin{pmatrix}6\\-1\end{pmatrix}$, $\overrightarrow{BC} = \begin{pmatrix}-2\\5\end{pmatrix}$, and $\overrightarrow{DC} = \begin{pmatrix}2\\-3\end{pmatrix}$.

Oct/Nov 2020

ORT makes up a triangle. $X$ lies on $TR$ so that $TX : XR = 3 : 2$. $O$ is the origin, $\overrightarrow{OR} = r$ and $\overrightarrow{OT} = t$. The diagram is labelled NOT TO SCALE.

Oct/Nov 2021

Find the value of $3\mathbf{q}$.

Oct/Nov 2022

The origin is $O$ at $(0,0)$, $A$ has coordinates $(8,1)$ and $B$ has coordinates $(2,5)$.

Oct/Nov 2023

Find $3\begin{pmatrix}6\\-4\end{pmatrix}$.

Oct/Nov 2024

The diagram indicates that $OBD$ and $ACD$ are straight lines. $O$ is the origin, the position vector of $A$ is $\mathbf a$ and the position vector of $B$ is $\mathbf b$. $\overrightarrow{BC}=\frac13\overrightarrow{OA}$. $M$ is located at the midpoint of $CD$.

Oct/Nov 2025

$\overrightarrow{DE}=\begin{pmatrix}3\\-4\end{pmatrix}$

Oct/Nov 2025

In the diagram, OA lies parallel to CB. The ratio OA : CB is 4 : 3. $\overrightarrow{OA}=a$ and $\overrightarrow{OB}=b$.

Oct/Nov 2025

ABCD forms a parallelogram. AB = m and AD = p. F lies on BC, with BF = 4FC. E lies on AD, and AE : ED = 1 : 2.

Oct/Nov 2025