Vector basics Worked Solutions — A-Level Maths

Read off the components

The top number is the amount in the $\mathbf{i}$ (horizontal) direction and the bottom number is the amount in the $\mathbf{j}$ (vertical) direction. Here that is $5$ across and $-3$ up.

Attach the unit vectors

We simply multiply each component by its unit vector $\mathbf{i}$ or $\mathbf{j}$. This is just the two ways of writing the same vector.

Tidy the signs

A $+(-3)$ is written more neatly as $-3$. That gives the finished $\mathbf{i},\mathbf{j}$ form.

Restate the question in your own words

Before starting, it helps to say clearly what the question wants. This keeps the working focused on the goal.

Identify each component

The number in front of $\mathbf{i}$ is the horizontal part and the number in front of $\mathbf{j}$ is the vertical part.

Stack them into a column

The horizontal part goes on top and the vertical part goes underneath. Column form is just a tidy way of listing the same two numbers.

Add the top components

To add vectors we add the matching components. First add the two horizontal ($\mathbf{i}$) numbers together.

Add the bottom components

Now add the two vertical ($\mathbf{j}$) numbers. Adding a negative is the same as subtracting.

Write the result

Put the two results back into a column. That is the sum of the vectors.

Subtract the top components

Subtracting vectors works component by component, just like adding. Start with the horizontal parts.

Subtract the bottom components

Now do the vertical parts. $2-8$ is negative because we take away more than we started with.

Write the result

Collect the two answers into a single column vector.

Multiply the top component

Multiplying a vector by a number means multiplying every component by that number. Start with the horizontal part.

Multiply the bottom component

Do the same to the vertical part. A positive times a negative gives a negative.

Write the result

The vector points the same way but is three times as long.