Vector geometry: past paper questions

Where the questions come from

• Question 25May/June 2026, paper 21Structured7 marksHas a diagram

  The diagram shows triangle $ABC$ and straight line $ACD$. $P$ lies on $AB$ and $AP:PB=4:1$. $Q$ lies on $BC$ and $BQ=\frac{2}{5}BC$.

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• Question 24Oct/Nov 2025, paper 21Structured4 marksHas a diagram

  In the diagram, $OBD$ and $ACD$ are straight lines. $O$ is the origin, the position vector of $A$ is $\mathbf{a}$ and the position vector of $B$ is $\mathbf{b}$. $\overrightarrow{BC}=\frac{1}{3}\overrightarrow{OA}$ $M$ is the midpoint of $CD$.

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• Question 26Oct/Nov 2023, paper 22Structured3 marks

  The diagram shows a parallelogram $OPQT$. The position vector of $P$ is $\mathbf{a}$ and the position vector of $T$ is $\mathbf{b}$. $K$ is on $PQ$ so that $PK:KQ=3:1$.

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• Question 22May/June 2026, paper 22Structured5 marksHas a diagram

  $OABC$ is a parallelogram. $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$. $P$ is a point on $AC$ and $AP:PC=1:4$.

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• Question 18Oct/Nov 2025, paper 42Structured4 marks

  In the diagram, $OA$ is parallel to $CB$. $OA:CB=4:3$ $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. Find $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

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• Question 10Oct/Nov 2023, paper 41Structured13 marksHas a diagram

  $ABC$ is a triangle. $B$ is the point $(1,-10)$, $A$ is the point $(4,14)$ and $\overrightarrow{CA}=\left(\begin{array}{c}-11\\ 8\end{array}\right)$. Find the coordinates of $C$.

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• Question 26Oct/Nov 2025, paper 43Structured5 marksHas a diagram

  $ABCD$ is a parallelogram. $\overrightarrow{AB}=\mathbf{m}$ and $\overrightarrow{AD}=\mathbf{p}$. $F$ is a point on $BC$ and $BF=4FC$.

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• Question 24May/June 2024, paper 22Structured5 marks

  $O$ is the origin and $OPQR$ is a parallelogram. $M$ is the midpoint of $PQ$ and $N$ divides $QR$ in the ratio $2:1$. $\overrightarrow{OP}=\mathbf{a}$ and $\overrightarrow{OR}=\mathbf{b}$.

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• Question 21May/June 2025, paper 21Structured5 marks

  $OABC$ is a rhombus and $O$ is the origin. The diagonals of the rhombus intersect at $P$. $\overrightarrow{OP}=2\mathbf{m}$ and $\overrightarrow{AP}=\mathbf{n}$.

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• Question 26May/June 2024, paper 23Structured4 marksHas a diagram

  In the diagram, $O$ is the origin. $\overrightarrow{OP}=\mathbf{p}$ and $\overrightarrow{OQ}=\mathbf{q}$. $R$ is the point of intersection of $PQ$ and $OS$, with $PR:RQ=1:2$ and $OR=RS$.

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• Question 9May/June 2025, paper 22Structured6 marks

  $A$ is the point $(3,-1)$. $$\overrightarrow{AB}=\left(\begin{array}{c}2\\ -4\end{array}\right)$$ $\overrightarrow{AC}=2\overrightarrow{AB}$ Find the coordinates of the point $C$.

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• Question 5May/June 2024, paper 41Structured17 marks

  $P$ is the point $(1,7)$. $Q$ is the point $(5,-5)$. Find $\overrightarrow{PQ}$.

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• Question 23May/June 2025, paper 22Structured3 marksHas a diagram

  In the diagram, $OA$ is 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 $\mathbf{b}$.

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