Vectors and position vectors

The idea

A vector in $\mathbf{i}, \mathbf{j}$ form, such as $3\mathbf{i} - 2\mathbf{j}$ , means $3$ across and $2$ down. The vector from $A$ to $B$ is $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ : end minus start.

Write each point as a position vector from the origin.
Subtract: $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$ .
Add or scale vectors component by component.

Watch out $\overrightarrow{AB}$ is $\mathbf{b} - \mathbf{a}$ , not $\mathbf{a} + \mathbf{b}$ .

See it worked

Worked example

$A$ is $(2, -1)$ and $B$ is $(5, 3)$ . Find $\overrightarrow{AB}$ .

$\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ .
$= (5\mathbf{i} + 3\mathbf{j}) - (2\mathbf{i} - \mathbf{j})$ .
$= 3\mathbf{i} + 4\mathbf{j}$ .

Answer $3\mathbf{i} + 4\mathbf{j}$

Your turn

Question 1Non-calculator3 marks

$\overrightarrow{OA} = 4\mathbf{i} - 2\mathbf{j}$ and $\overrightarrow{OB} = \mathbf{i} + 5\mathbf{j}$ . Find $\overrightarrow{AB}$ . Give the $\mathbf{i}$ and $\mathbf{j}$ components.

Question 2Non-calculator3 marks

$\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}$ and $\mathbf{b} = \mathbf{i} - 4\mathbf{j}$ . Find $3\mathbf{a} - 2\mathbf{b}$ . Give the $\mathbf{i}$ and $\mathbf{j}$ components.

Question 3Non-calculator3 marks

$A$ is $(1, 7)$ and $B$ is $(5, -3)$ . Find the position vector of the midpoint of $AB$ . Give the $\mathbf{i}$ and $\mathbf{j}$ components.

In past papers

39 official questions in 36 papers since 2018. The latest:

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