Position & Displacement Vectors

What is a Position Vector?

A position vector tells us the location of a point relative to a fixed origin, usually called $O$.

The position vector of a point $A$ is written as: $\vec{a} = \overrightarrow{OA}$

The components of the vector are just the coordinates of the point.

Example:

If $A$ is at $(4, -3)$, then:

$$\vec{a} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$$

What is a Displacement Vector?

A displacement vector shows how to get from one point to another.

The displacement from point $A$ to point $B$ is written as: $$\overrightarrow{AB}$$

If $\vec{a} = \overrightarrow{OA}$ and $\vec{b} = \overrightarrow{OB}$, then:

$$\overrightarrow{AB} = \vec{b} - \vec{a}$$

This formula works because going from $A$ to $B$ is the same as going back to the origin from $A$, then from the origin to $B$.

 
Example

Given:

Point $P$ has position vector: $$\vec{p} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$

Point $Q$ has position vector: $$\vec{q} = \begin{pmatrix} 6 \\ -10 \end{pmatrix}$$

Find:

$$\overrightarrow{PQ}$$

Step 1: Use the rule

$$\overrightarrow{PQ} = \vec{q} - \vec{p}$$

Step 2: Subtract the vectors

$$\begin{pmatrix} 6 \\ -10 \end{pmatrix} - \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -12 \end{pmatrix}$$

So,

$$\overrightarrow{PQ} = \begin{pmatrix} 3 \\ -12 \end{pmatrix}$$