Vector geometry

$\overrightarrow{XY} = 3a + 2b$ together with $\overrightarrow{ZY} = 6a + 4b$.

Feb/March 2020

The figure depicts triangle $OAB$. $C$ is placed at the midpoint of $OA$. $OC = m$ and $CB = n$. $E$ is situated on $AB$ and $AE : EB = 4 : 5$. The figure is marked NOT TO SCALE.

Feb/March 2025

$OAPB$ forms a parallelogram. $O$ serves as the origin. $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $M$ is the midpoint of $BP$.

May/June 2015

The diagram depicts quadrilateral $GHIJK$. The side vectors are labelled as $\overrightarrow{GH}=a$, $\overrightarrow{HJ}=b$, and $\overrightarrow{KJ}=c$. Point $L$ is located on $GK$ with $LK=3GL$. The diagram is labelled NOT TO SCALE.

May/June 2016

$OPQR$ forms a rectangle, with $O$ at the origin. $M$ lies halfway along $RQ$, and $PT:TQ=2:1$. $\overrightarrow{OP}=\mathbf{p}$ and $\overrightarrow{OR}=\mathbf{r}$.

May/June 2016

The diagram depicts triangle $OPQ$. $O$ is the origin, $\vec{OP}=\vec{p}$ and $\vec{OQ}=\vec{q}$. $Z$ lies on $PQ$ so that $PZ:ZQ = 5:2$. The diagram is not drawn to scale.

May/June 2017

A triangle diagram, drawn not to scale, marks points $O$, $P$ and $Q$. $O$ is the origin. The vector $\vec{OP}$ is $\mathbf{p}$, while $\vec{OQ}$ is $\mathbf{q}$. Point $T$ is on $QP$ so that $QT:TP=2:1$.

May/June 2018

The diagram depicts parallelogram $ABCD$, where $\overrightarrow{AB}=q$ and $\overrightarrow{AD}=p$. $ABM$ is collinear, with $AB:BM = 1:1$. $ADN$ is collinear, with $AD:DN = 3:2$.

May/June 2019

The figure shows triangle $O$, $P$ and $Q$. Vector $\vec{OP}=a$ and vector $\vec{OQ}=b$. Point $S$ lies on $PQ$ so that $PS:SQ = 4:5$. The diagram is not drawn to scale.

May/June 2021

A diagram includes the points $O$, $A$, $D$ and $B$. The diagram is not drawn to scale. $\vec{OA} = x$, $\vec{OB} = y$ and $\vec{OD} = \frac{3}{7}x + \frac{4}{7}y$.

May/June 2022

O is the origin, and OPQR forms a parallelogram. M lies halfway along PQ, while N divides QR in the ratio $2 : 1$. $\vec{OP}=\mathbf{a}$ and $\vec{OR}=\mathbf{b}$.

May/June 2024

In the diagram, $O$ marks the origin. $\overrightarrow{OP} = \mathbf{p}$ and $\overrightarrow{OQ} = \mathbf{q}$. $R$ is where $PQ$ meets $OS$, with $PR : RQ = 1 : 2$ and $OR = RS$. The diagram is not drawn to scale.

May/June 2024

Find $\vec{PQ}$.

May/June 2024

In the diagram, $OA$ runs parallel to $BC$. $BC = 3OA$. $M$ is the midpoint of $AC$. The position vector of $A$ is $\mathbf{a}$ and the position vector of $B$ is $\mathbf{b}$.

May/June 2025

A quadrilateral $OXPY$ is shown, and its diagonals intersect at $T$. The vector $\overrightarrow{OX}$ is $3a$, while $\overrightarrow{OY}$ is $12b$. The upper side $YP$ has the label $5a - 2b$. The left side is marked $12b$, and the bottom side $OX$ is marked $3a$. The diagram is NOT TO SCALE.

May/June 2025

The diagram represents triangle $ABC$. $\overrightarrow{BC} = a$ and $\overrightarrow{AC} = b$.

Oct/Nov 2015

Find the value of $u$ and of $v$.

Oct/Nov 2016

Determine the coordinates of point $E$.

Oct/Nov 2017

In the figure, $OABC$ forms a parallelogram. The lines $OP$ and $CA$ meet at $X$, and $CP:PB = 2:1$. Also, $OA = \vec{a}$ and $OC = \vec{c}$. The sketch is not drawn to scale and uses arrows to show the vectors $\vec{a}$ along $OA$ and $\vec{c}$ along $OC$, with $X$ marked where the diagonals meet.

Oct/Nov 2018

A triangle is drawn and it is NOT TO SCALE. $O$ denotes the origin, $\overrightarrow{OP} = 2\overrightarrow{OA}$, $\overrightarrow{OQ} = 3\overrightarrow{OB}$ and $\overrightarrow{PM} = \overrightarrow{MQ}$. $\overrightarrow{OP} = \mathbf{p}$ and $\overrightarrow{OQ} = \mathbf{q}$.

Oct/Nov 2019

The diagram depicts a parallelogram $CDEF$. $\overrightarrow{FE} = m$ and $\overrightarrow{CE} = n$. $B$ lies at the midpoint of $CD$. $\overrightarrow{FA} = 2\overrightarrow{AC}$.

Oct/Nov 2020

The figure contains triangle $OAB$ together with the straight line $OAC$. The ratio $OA : OC = 2 : 5$, and $M$ is the midpoint of $AB$. Also, $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$. The diagram is not drawn to scale.

Oct/Nov 2020

Find the coordinates for point $G$.

Oct/Nov 2021

For $\left|\left(\begin{array}{c}9m \\ 40m\end{array}\right)\right| = \frac{205}{2}$, determine the two possible values of $m$.

Oct/Nov 2022

The figure depicts a parallelogram $OPQT$. The position vector of $P$ is $\mathbf{a}$ and the position vector of $T$ is $\mathbf{b}$. $K$ lies on $PQ$ such that $PK : KQ = 3 : 1$. The straight lines $OK$ and $TQ$ are produced until they intersect at $X$. The sketch is marked NOT TO SCALE.

Oct/Nov 2023

Triangle ABC is given, with B at $(1,-10)$, A at $(4,14)$ and $\overrightarrow{CA} = \begin{pmatrix}-11\\8\end{pmatrix}$.

Oct/Nov 2023

In the diagram, $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$. The ratio $AK : KB = 2 : 1$. Also, $OK = KC$. The diagram is marked "NOT TO SCALE".

Oct/Nov 2024

Work out $2 \begin{pmatrix}3\\-5\end{pmatrix} - \begin{pmatrix}2\\-7\end{pmatrix}$.

Oct/Nov 2024