Vector Notation and Operations Revision Notes | GCSE AQA Higher Maths

1. What Is a Vector and Why Does It Matter?

Imagine giving someone directions: "walk 3 metres" tells them how far, but not where to go. "Walk 3 metres east" is far more useful — that combination of distance and direction is exactly what a vector captures. Scalars like temperature or mass only need a number; vectors need a number and a direction.

Vectors appear throughout GCSE Maths in geometry and transformations. On the Higher-tier paper you will be asked to add, subtract, and scale vectors, and to represent them both as arrows on a diagram and as column vectors. This lesson builds every one of those skills from the ground up.

The contrast with a scalar is worth fixing in your mind now. Speed is a scalar (50 km/h); velocity is a vector (50 km/h due north). Mass is a scalar; displacement is a vector. Whenever direction matters, you are dealing with a vector.

2. Representing Vectors: Diagrams and Column Notation

There are two ways to write a vector that you need to be comfortable switching between. The diagrammatic representation is an arrow on a grid: the arrow points in the direction of travel and its length is proportional to the magnitude. The column vector packages the same information into two numbers stacked vertically.

Think of the column vector like a set of instructions for a robot: "move $x$ steps right, then $y$ steps up." A negative $x$ means move left; a negative $y$ means move down. The order always stays the same — horizontal on top, vertical on the bottom.

That last point is crucial for the exam: two arrows that are parallel, the same length, and point in the same direction represent identical vectors, regardless of where they are drawn on the grid. Position does not define a vector — displacement does.

3. Adding Vectors

Now that you can represent a vector, the next question is: what happens when you combine two of them? Adding vectors is the mathematical way of describing a journey made in two stages. If you walk from $A$ to $B$ and then from $B$ to $C$, the overall displacement from $A$ to $C$ is the resultant vector.

The rule for column vectors is refreshingly direct: add the top numbers together and add the bottom numbers together. Horizontal and vertical components are kept completely separate throughout the calculation.

Worked example: Adding Two Vectors

Vector $\mathbf{p} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$ and vector $\mathbf{q} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}$. Find $\mathbf{p} + \mathbf{q}$.

1. $\mathbf{p} + \mathbf{q} = \begin{pmatrix} 4 \\ -1 \end{pmatrix} + \begin{pmatrix} -2 \\ 5 \end{pmatrix}$ — Set up the calculation by writing both column vectors side by side. [M1]
2. $4 + (-2) = 2$ — Top numbers only. Adding a negative is the same as subtracting.
3. $-1 + 5 = 4$ — Bottom numbers only. Keep horizontal and vertical completely separate.
4. $\mathbf{p} + \mathbf{q} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$ — Combine the two results into a single column vector. [A1]

Answer: $\mathbf{p} + \mathbf{q} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$

Diagrammatically, vector addition follows the tip-to-tail rule: draw the first vector, then start the second vector from the tip (arrowhead) of the first. The resultant goes from the tail of the first vector to the tip of the second. This is sometimes called the triangle law of vector addition.

4. Subtracting Vectors

Subtraction follows the same component-by-component logic, but there is a geometric insight worth understanding first. Subtracting $\mathbf{b}$ from $\mathbf{a}$ is the same as adding the negative of $\mathbf{b}$. The negative of a vector has the same magnitude but points in exactly the opposite direction.

Worked example: Subtracting Vectors

Given $\mathbf{m} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$ and $\mathbf{n} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$, find $\mathbf{m} - \mathbf{n}$.

1. $\mathbf{m} - \mathbf{n} = \begin{pmatrix} 6 \\ 2 \end{pmatrix} - \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ — Write both vectors clearly before computing. [M1]
2. $6 - 3 = 3$ — Top number of first vector minus top number of second vector.
3. $2 - (-1) = 2 + 1 = 3$ — Subtracting a negative flips the sign. This is where sign errors most commonly occur — watch carefully.
4. $\mathbf{m} - \mathbf{n} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}$ — Combine into a single column vector. [A1]

Answer: $\mathbf{m} - \mathbf{n} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}$

Notice step 3 carefully: $2 - (-1)$ becomes $2 + 1 = 3$. Subtracting a negative is one of the most common sources of errors in vector questions. Whenever you see a negative component in the vector being subtracted, pause and rewrite it as an addition before computing.

5. Multiplying a Vector by a Scalar

Earlier we established that a scalar is a plain number with no direction. When you multiply a vector by a scalar, you scale every component by that number. The direction stays the same (or reverses if the scalar is negative), but the magnitude changes proportionally.

Worked example: Multiplying a Vector by a Scalar

Vector $\mathbf{v} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$. Find (a) $3\mathbf{v}$ and (b) $-2\mathbf{v}$.

1. $3\mathbf{v} = 3 \times \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 3 \times 2 \\ 3 \times (-3) \end{pmatrix} = \begin{pmatrix} 6 \\ -9 \end{pmatrix}$ — Every component is multiplied by 3. The vector is three times as long and points in the same direction. [A1]
2. $-2\mathbf{v} = -2 \times \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} -2 \times 2 \\ -2 \times (-3) \end{pmatrix} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}$ — A negative scalar reverses the direction. $-2 \times (-3) = 6$ — two negatives multiply to give a positive. [A1]

Answer: (a) $3\mathbf{v} = \begin{pmatrix} 6 \\ -9 \end{pmatrix}$, (b) $-2\mathbf{v} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}$

To effectively learn vector operations, practicing through structured problems will aid retention. Work through each problem before checking the solution — the attempt matters as much as the answer for building fluency.

6. Combining Operations: Multi-Step Vector Problems

GCSE Higher exam questions rarely test just one operation in isolation. The real skill is chaining addition, subtraction, and scalar multiplication together in a single calculation. This is where the earlier building blocks pay off.

A common question type asks you to find $\overrightarrow{AC}$ given $\overrightarrow{AB}$ and $\overrightarrow{BC}$, or to express one vector in terms of others. The strategy is always the same: plan a route, write the vectors along that route, then add or subtract as needed.

Worked example: Multi-Step Vector Calculation

Given $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$, find $2\mathbf{a} + 3\mathbf{b}$.

1. $2\mathbf{a} = 2 \times \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$ — $2 \times 3 = 6$ and $2 \times 1 = 2$. [M1]
2. $3\mathbf{b} = 3 \times \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} -3 \\ 12 \end{pmatrix}$ — $3 \times (-1) = -3$ and $3 \times 4 = 12$. [M1]
3. $2\mathbf{a} + 3\mathbf{b} = \begin{pmatrix} 6 \\ 2 \end{pmatrix} + \begin{pmatrix} -3 \\ 12 \end{pmatrix} = \begin{pmatrix} 6 + (-3) \\ 2 + 12 \end{pmatrix} = \begin{pmatrix} 3 \\ 14 \end{pmatrix}$ — Add corresponding components. $6 + (-3) = 3$ and $2 + 12 = 14$. [A1]

Answer: $2\mathbf{a} + 3\mathbf{b} = \begin{pmatrix} 3 \\ 14 \end{pmatrix}$

7. Deliberate Practice

This lesson covers vector operations comprehensively, ensuring you can manage addition, subtraction, and scalar multiplication confidently. In addition to conceptual understanding, work through structured practice problems that mirror exam scenarios to solidify your skills.

8. Exam Technique and Common Mistakes

7. Deliberate Practice