​Position vectors and displacement vectors

​​In a nutshell

Vectors are used to describe a position of a point in space. A position vector gives the position of a point relative to a fixed origin, $$O$$. A displacement vector describes movement from one position to another.

Position vectors

A position vector gives the position of a point relative to a fixed origin.

Point $$P$$ has coordinates $$(a,b)$$. The position vector of $$P$$ from the fixed origin $$O$$ to $$P$$, is $$\overrightarrow{OP}$$.​

​$$\boxed{\overrightarrow{OP}=a\textbf{i}+b\textbf{j}=\begin{pmatrix}a\\b\end{pmatrix}}$$​​

Displacement vectors

A displacement vector describes movement from one position to another. In a two-coordinate system, the difference between the position vectors of the final point and the initial point, relative to the origin, gives your displacement.

If point $$A$$ is the initial position, and point $$B$$ is the final position, the displacement vector $$\overrightarrow{AB}$$ is the difference between the two points.

The position vector of $$A$$ is $$\overrightarrow{OA}=\textbf{a}$$.

The position vector of $$B$$ is $$\overrightarrow{OB}=\textbf{b}$$.

$$\boxed{\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}}$$​​

Example 1

The diagram shows points $$A$$ and $$B$$. Find, in column vector form, the position vector of $$A$$, the position vector of $$B$$, and the vector $$\overrightarrow{AB}$$.

The position vector of $$A$$ is its position relative to the origin, $$O$$:

​$$\underline{\overrightarrow{OA}=\begin{pmatrix}-1\\2\end{pmatrix}}$$​​

The position vector of $$B$$ is its positio​n relative to the origin, $$O$$:

​$$\underline{\overrightarrow{OB}=\begin{pmatrix}5\\2\end{pmatrix}}$$​​

$$\overrightarrow{AB}$$ is the displacement vector between t​he two points:

​$$\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\\&=\begin{pmatrix}5\\2\end{pmatrix}-\begin{pmatrix}-1\\2\\\end{pmatrix}\\&=\underline{\begin{pmatrix}6\\0\end{pmatrix}}\end{aligned}$$​​

Example 2

Points $$A$$ and $$B$$ have position vectors $$4\textbf{i}+3\textbf{j}$$ and $$-3\textbf{i}+\textbf{j}$$ respectively. Find $$\overrightarrow{AB}$$.

$$\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\\&=(-3\textbf{i}+\textbf{j})-(4\textbf{i}+3\textbf{j})\\&=\underline{-7\textbf{i}-2\textbf{j}}\end{aligned}$$

Example 3

Point $$A$$ has coordinates $$(3,2)$$ and $$\overrightarrow{AB}=-2\textbf{i}-8\textbf{j}$$. Find the position vector of $$B$$ and its exact magnitude.

The position vector of $$A$$ is:

$$\overrightarrow{OA}=3\textbf{i}+2\textbf{j}$$

We know that $$\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\end{aligned}$$. So:

$$\begin{aligned}\overrightarrow{AB}&=\overrightarrow{OB}-\overrightarrow{OA}\\\overrightarrow{OB}&=\overrightarrow{AB}+\overrightarrow{OA}\\&=(-2\textbf{i}-8\textbf{j})+(3\textbf{i}+2\textbf{j})\\&=\underline{\textbf{i}-6\textbf{j}}\end{aligned}$$​​

​​

For the magnitude of the position vector of $$B$$:

$$\begin{aligned}|\overrightarrow{OB}|&=\sqrt{1^2+(-6)^2}\\&=\sqrt{1+36}\\&=\underline{\sqrt{37}}\end{aligned}$$​​

Note: You have been asked for the exact value, so leave your answer in surd form.