22.3 Vectors in geometric problems and proofs — Practise

H In triangle $ABC$ , $\overrightarrow{AB}=\mathbf{a}$ and $\overrightarrow{BC}=\mathbf{b}$ . Find $\overrightarrow{AC}$ .
H If $\overrightarrow{AB}=\mathbf{u}+\mathbf{v}$ , write $\overrightarrow{BA}$ .
H In parallelogram $ABCD$ , $\overrightarrow{AB}=\mathbf{u}$ and $\overrightarrow{AD}=\mathbf{v}$ . Find $\overrightarrow{AC}$ .
H In the same parallelogram, find $\overrightarrow{DB}$ .
H A segment has endpoint position vectors $\mathbf{p}$ and $\mathbf{q}$ . Write the midpoint position vector.
H In parallelogram $ABCD$ , take $A$ as the origin, with $B=\mathbf{u}$ and $D=\mathbf{v}$ . Show the diagonals have the same midpoint.
H In triangle $OAB$ , the position vectors of $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ . Point $M$ is the midpoint of $AB$ . Find $\overrightarrow{OM}$ .
H A quadrilateral has successive side vectors $\mathbf{p}$ , $\mathbf{q}$ , $-\mathbf{p}$ and $-\mathbf{q}$ . Explain why it closes.
H Point $P$ is the midpoint of $AB$ and point $Q$ is the midpoint of $BC$ , where $\overrightarrow{AB}=\mathbf{u}$ and $\overrightarrow{BC}=\mathbf{v}$ . Which method finds $\overrightarrow{PQ}$ : add the route vectors through $B$ , or halve the diagonal $AC$ ? Give a reason, then find it.

Answers and concise solutions

1. Vector AC

Answer: $\overrightarrow{AC}=\mathbf{a}+\mathbf{b}$ , by adding the two route vectors.

2. Reverse vector

Answer: $\overrightarrow{BA}=-\mathbf{u}-\mathbf{v}$ .

3. Diagonal AC

Answer: $\overrightarrow{AC}=\mathbf{u}+\mathbf{v}$ .

4. Diagonal DB

Answer: $\overrightarrow{DB}=\mathbf{u}-\mathbf{v}$ , using $D$ to $A$ to $B$ .

5. Midpoint position vector

Answer: $\frac12(\mathbf{p}+\mathbf{q})$ .

6. Midpoint equality

Answer: $C=\mathbf{u}+\mathbf{v}$ . Midpoint of $AC=\frac12(\mathbf{u}+\mathbf{v})$ ; midpoint of $BD=\frac12(\mathbf{u}+\mathbf{v})$ . The midpoints are equal, so the diagonals bisect each other.

7. Vector OM

Answer: $\overrightarrow{OM}=\frac12(\mathbf{a}+\mathbf{b})$ .

8. Closure

Answer: The total vector is $\mathbf{p}+\mathbf{q}-\mathbf{p}-\mathbf{q}=\mathbf{0}$ , so the route returns to its start.

9. Method choice: route or diagonal

Answer: Add the route vectors through $B$ : $\overrightarrow{PQ}=\overrightarrow{PB}+\overrightarrow{BQ}=-\frac12\mathbf{u}+\frac12\mathbf{v}=\frac12(\mathbf{v}-\mathbf{u})$ . Halving the diagonal would give the midpoint of $AC$ , not $PQ$ .