Vectors: proving lines are parallel

Vectors

(1MA1) · Higher

ORIGINAL

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

Board and spec code confirmed against Edexcel 1MA1 · registry checked 2026-07-09How this checking works

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 $PQ:AB=2:3$ .