A-level · Vectors

Position vectorsExam-style questions

Pinning points to the origin with position vectors, the "destination minus start" rule AB = b − a, the distance between two points as |b − a|, the midpoint as ½(a + b), and finding a point that divides a segment in a ratio.

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15 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The points $A$ and $B$ have position vectors $\begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 6 \\ 1 \\ -2 \end{pmatrix}$ respectively.
(a) Find the exact length of $AB$ .

(2 marks)
(b) Find the position vector of the midpoint of $AB$ .

(1 mark)
Level 2 · 3 marksTry it
2
Relative to an origin $O$ , points $A$ and $B$ have position vectors $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $4\mathbf{i} - \mathbf{j} + 5\mathbf{k}$ . Find $\vec{AB}$ .
Level 2 · 2 marksTry it
3
Relative to a fixed origin $O$ $O$
- the point $A$ has position vector $\mathbf{i} - 4\mathbf{j} + 6\mathbf{k}$
- the point $B$ has position vector $-3\mathbf{i} + 2\mathbf{j} + a\mathbf{k}$
where $a$ $a$ is a positive integer.
(a) Show that $\left|\overrightarrow{OA}\right| = \sqrt{53}$ $O A = 53$

(1 mark)
(b) Find the least value of $a$ $a$ such that $\left|\overrightarrow{OB}\right| > \left|\overrightarrow{OA}\right|$ $O B > O A$

(2 marks)
Level 2 · 3 marksTry it
4
Relative to a fixed origin $O$ $O$
- the point $A$ has position vector $5\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$
- the point $B$ has position vector $-\mathbf{i} + 8\mathbf{j} + 2\mathbf{k}$
(a) Find $\overrightarrow{AB}$ $A B$

(1 mark)
(b) Find $\left|\overrightarrow{AB}\right|$ $A B$ , writing your answer as a fully simplified surd.

(2 marks)
Level 2 · 3 marksTry it
5
Points $A$ and $B$ have position vectors $2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $5\mathbf{i} + 3\mathbf{j} - \mathbf{k}$ . Find the exact distance $AB$ .
Level 3 · 3 marksTry it

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