Vector proofs

Higher only

The idea

Write any vector as a route through known vectors. Two vectors are parallel when one is a multiple of the other; three points lie on a straight line when the vectors between them are parallel and share a point.

Go from start to finish along known vectors, adding them.
Going backwards along a vector subtracts it.
To prove parallel, show one vector is a number times the other.

Watch out Going against the arrow subtracts the vector.

See it worked

Worked exampleNon-calculator3 marks

The diagram shows triangle $OAB$ .

$\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .

$M$ is the midpoint of $AB$ .

(a) Find $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ .

(b) Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ . Give your answer in its simplest form.

(a) $\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} = -\mathbf{a} + \mathbf{b}$ .
(b) $\overrightarrow{OM} = \overrightarrow{OA} + \tfrac{1}{2}\overrightarrow{AB} = \mathbf{a} + \tfrac{1}{2}(\mathbf{b} - \mathbf{a})$
$= \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b}$ .

Answer (a) $\mathbf{b} - \mathbf{a}$ ; (b) $\frac{1}{2}(\mathbf{a} + \mathbf{b})$

Your turn

Question 1Non-calculator2 marks

The diagram shows triangle $OAB$ .

$\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .

$M$ is the midpoint of $AB$ . $P$ lies on $OB$ with $OP : PB = 2 : 3$ .

Find $\overrightarrow{PM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ .

Question 2Non-calculator4 marks

The diagram shows triangle $OAB$ .

$\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .

$M$ lies on $OA$ with $OM : MA = 1 : 2$ . $N$ lies on $AB$ with $AN : NB = 2 : 1$ .

The lines $BM$ and $ON$ intersect at $X$ .

Work out the fraction $\dfrac{OX}{ON}$ .

You must show your working using vectors.

Question 3Non-calculator4 marks

The diagram shows triangle $ABC$ .

$M$ is the midpoint of $AB$ . $N$ lies on $AC$ with $AN : NC = 1 : 2$ .

The line $MN$ meets the line $CB$ extended at $P$ .

$\overrightarrow{BP} = k\overrightarrow{BC}$

Work out the value of $k$ .

You must show your working using vectors.

In past papers

27 official questions in 27 papers since 2017. The latest:

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