11.4PureCore

Position Vectors

A position vector pins a point to the origin. Once points have position vectors, the vector joining any two of them follows from one simple rule — the key to all the geometry problems in the next lesson.

25 min Vectors

Edexcel AS Level Maths: 11.4 Position VectorsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Understand what a position vector is
Write the position vector of a point
Find the vector between two points
Find the distance between two points using vectors

Position vectors

The $\textbf{position vector}$ of a point $A$ is $\overrightarrow{OA}$ , the vector from the origin $O$ to $A$ . It is usually written $\mathbf{a}$ . So the point $(3, 5)$ has position vector $\begin{pmatrix} 3 \\ 5 \end{pmatrix}$ .

The vector joining two points

To get from $A$ to $B$ , go back to the origin then out to $B$ : $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ . Always “end minus start”.

\overrightarrow{AB} = \mathbf{b} - \mathbf{a}

End point’s position vector minus the start point’s.

Example 1Find

\overrightarrow{AB}

A(1, 2)

and

B(4, 7)

\overrightarrow{AB} = \begin{pmatrix} 4 \\ 7 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix}

Answer

\begin{pmatrix} 3 \\ 5 \end{pmatrix}

Tip — AB = b − a, not a − b. Getting the order wrong reverses the vector.

Distance between points

The distance from $A$ to $B$ is the magnitude of $\overrightarrow{AB}$ — exactly the distance formula in disguise.

|\overrightarrow{AB}| = |\mathbf{b} - \mathbf{a}|

Find the joining vector, then its magnitude.

Example 1Find the distance from

A(1, 2)

B(4, 6)

\overrightarrow{AB} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}

2Magnitude

= \sqrt{9 + 16} = 5

Answer

5

Formula recap

\overrightarrow{OA} = \mathbf{a}

Position vector of A.

\overrightarrow{AB} = \mathbf{b} - \mathbf{a}

Vector joining two points.

|\overrightarrow{AB}| = |\mathbf{b} - \mathbf{a}|

Distance between points.

Common mistakes to avoid

Writing AB = a − b.

It is end minus start: AB = b − a.

Confusing a position vector with a displacement between two general points.

A position vector is always measured from the origin O.

Key takeaways

A position vector OA = a runs from the origin to the point.
AB = b − a (end minus start).
Distance from A to B is |b − a|.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Position Vectors. Use the feedback to decide what needs another look.

11 past-paper questions · 22 marks

Practise this topic

Related lessons

All 6 lessons

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