Vectors: proving lines are parallel

Vectors

(1MA1) · Higher

ORIGINAL

Express two routes with vectors, then prove parallel lines by finding a positive scalar multiple.

Verified against Edexcel 1MA1 (2026 spec)

Question

In triangle $OAB$ , $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ . Point $P$ lies on $OA$ with $OP:PA=2:1$ , and point $Q$ lies on $OB$ with $OQ:QB=2:1$ . Prove that $PQ$ is parallel to $AB$ , and state the ratio $PQ:AB$ .

Every step worked, with the reasoning.

1
$\overrightarrow{OP}=\dfrac{2}{3}\mathbf{a}$ and $\overrightarrow{OQ}=\dfrac{2}{3}\mathbf{b}$
Each point is two-thirds of the way from $O$ to the other vertex.
2
$\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}$
Use the route $P \to O \to Q$ .
3
$\overrightarrow{PQ}=\dfrac{2}{3}\mathbf{b}-\dfrac{2}{3}\mathbf{a}=\dfrac{2}{3}(\mathbf{b}-\mathbf{a})$
Substitute the two position vectors and factorise.
4
$\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$
Use the route $A \to O \to B$ .
5
$\overrightarrow{PQ}=\dfrac{2}{3}\overrightarrow{AB}$
A positive scalar multiple has the same direction, so $PQ$ is parallel to $AB$ and its length is two-thirds as large.

Answer: $PQ \parallel AB$ and $PQ:AB=2:3$ .

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