22.3 Vectors in geometric problems and proofs — Practise with help

Warm-up: remember

Work out $\begin{pmatrix}2\\-1\end{pmatrix}+\begin{pmatrix}-3\\4\end{pmatrix}$ .
What vector reverses $\begin{pmatrix}5\\-2\end{pmatrix}$ ?

Answers and feedback

$\begin{pmatrix}-1\\3\end{pmatrix}$ ; add matching components.
$\begin{pmatrix}-5\\2\end{pmatrix}$ ; reversing a vector changes both signs.

Practise with help

1. In triangle $PQR$ , $\overrightarrow{PQ}=\mathbf{a}$ and $\overrightarrow{QR}=\mathbf{b}$ . Find $\overrightarrow{PR}$ .

Hint

Follow the route from $P$ to $Q$ to $R$ .

Next step

$\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}$ .

Full working

$\overrightarrow{PR}=\mathbf{a}+\mathbf{b}$ .

2. In parallelogram $ABCD$ , $\overrightarrow{AB}=\mathbf{u}$ and $\overrightarrow{AD}=\mathbf{v}$ . Find $\overrightarrow{AC}$ .

Hint

Use the route $A$ to $B$ to $C$ .

Next step

$\overrightarrow{BC}=\overrightarrow{AD}=\mathbf{v}$ .

Full working

$\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=\mathbf{u}+\mathbf{v}$ .

3. In the same parallelogram, find $\overrightarrow{DB}$ .

Hint

Use the route $D$ to $A$ to $B$ .

Next step

$\overrightarrow{DA}=-\mathbf{v}$ and $\overrightarrow{AB}=\mathbf{u}$ .

Full working

$\overrightarrow{DB}=-\mathbf{v}+\mathbf{u}=\mathbf{u}-\mathbf{v}$ .

4. A point $M$ is the midpoint of a segment with endpoint position vectors $\mathbf{p}$ and $\mathbf{q}$ . Write the position vector of $M$ .

Hint

A midpoint is halfway between the two endpoints.

Next step

Add the endpoint vectors, then multiply by $\frac12$ .

Full working

$\overrightarrow{OM}=\frac12(\mathbf{p}+\mathbf{q})$ .

5. Use position vectors to show that the diagonals of a parallelogram bisect each other.

Hint

Set one vertex at the origin and label the two side vectors $\mathbf{u}$ and $\mathbf{v}$ .

Next step

The other vertices have position vectors $\mathbf{u}$ , $\mathbf{v}$ and $\mathbf{u}+\mathbf{v}$ .

Full working

Both diagonal midpoints have position vector $\frac12(\mathbf{u}+\mathbf{v})$ . Therefore they are the same point and the diagonals bisect each other.

6. Explain why a vector proof can use any chosen origin.

Hint

Think about whether the final equality depends on the labels or on the point locations.

Next step

A different origin changes all position vectors by the same displacement.

Full working

The proof compares positions using consistent vectors. Choosing a different origin shifts each position vector equally, so the geometric equality remains true.