Position & Displacement Vectors

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Position & displacement vectors

A position vector describes where a specific point, A , is, relative to a fixed origin, O
- Lower-case bold (or underlined) letters are used
  - The point A has position vector a = $\vec{O A}$
Their components are equal to their coordinates
- The point with coordinates (3, -2) has position vector $(\begin{matrix} 3 \\ - 2 \end{matrix})$ from the origin

A displacement vector describes the direction and distance between two points
- The displacement vector from A to B is $\vec{A B}$
  - How to get from A to B
If the points A and B have position vectors a and b relative to O
- then A to B is the same as A to O (-a) followed by O to B (b)
  - $\vec{A B} = - a + b = b - a$
  - This is a useful rule to remember

Worked example

The points $P$ and $Q$ have position vectors $(\begin{matrix} 3 \\ 2 \end{matrix})$ and $(\begin{matrix} 6 \\ - 10 \end{matrix})$ respectively.

Find and simplify the vector $\vec{P Q}$ .

Answer:

Let $p$ and $q$ be position vectors of P and Q
$\vec{P Q}$ is the displacement vector from P to Q
Use the rule that $\vec{A B} = - a + b = b - a$

$\vec{P Q} = q - p$

Substitute in $p$ and $q$

$\vec{P Q} = (\begin{matrix} 6 \\ - 10 \end{matrix}) - (\begin{matrix} 3 \\ 2 \end{matrix})$

Expand and simplify

$\begin{array}{rcl} = & (\begin{matrix} 6 - 3 \\ - 10 - 2 \end{matrix}) \\ = & (\begin{matrix} 3 \\ - 12 \end{matrix}) \end{array}$

$\vec{P Q} = (\begin{matrix} 3 \\ - 12 \end{matrix})$