GB-34-05-001 · Vector geometry proof

In triangle OAB, $\overrightarrow{OA}=4\mathbf a$ and $\overrightarrow{OB}=\mathbf b$ . P lies on OA and Q lies on OB such that OP/OA = OQ/OB = $\frac{1}{3}$ . Use vectors to prove that PQ is parallel to AB.

Vectors: Parallel Lines Proof in a Triangle Booklet p. 90

Method

1) Find AB = AO + OB

2) Find PQ = PO + OQ

3) Factorise PQ to show PQ = k(AB) ("PQ is a multiple of AB ⇒ parallel")

Worked Example (GB-34-05-001)

OP =

\frac{4}{3}\mathbf a

and OQ =

\frac{1}{3}\mathbf b

Subtract OP from OQ and factor out

\frac{1}{3}

PQ is a non-zero scalar multiple of AB, so parallel.

Ans =

\overrightarrow{PQ}=\frac{1}{3}(\mathbf b-4\mathbf a)=\frac{1}{3}\overrightarrow{AB}