A Level Mathematics: Vectors Revision Made Simple

Quick answer

Vectors in A Level Maths often seem daunting, but they're just about direction and magnitude. Focus on understanding basic operations and applications rather than memorising. This way, you can tackle any vector question with confidence.

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What you need to know

A vector is a quantity that has both direction and magnitude. Think of it like an arrow pointing somewhere — its length is the magnitude, and where it points is the direction. In exams, you'll often deal with vectors in 2 D or 3 D space, represented as $\mathbf{𝑎} = \begin{pmatrix} 𝑎_1 \\ 𝑎_2 \end{pmatrix}$ or $\mathbf{𝑎} = \begin{pmatrix} 𝑎_1 \\ 𝑎_2 \\ 𝑎_3 \end{pmatrix}$.

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Vector Basics

Addition and Subtraction

When you add or subtract vectors, you do it component-wise.

Step 1: Combine the corresponding components. If $\mathbf{𝑎} = \begin{pmatrix} 𝑎_1 \\ 𝑎_2 \end{pmatrix}$ and $\mathbf{𝑏} = \begin{pmatrix} 𝑏_1 \\ 𝑏_2 \end{pmatrix}$, then $\mathbf{𝑎} + \mathbf{𝑏} = \begin{pmatrix} 𝑎_1 + 𝑏_1 \\ 𝑎_2 + 𝑏_2 \end{pmatrix}$.
Why: This keeps the vector's direction and magnitude consistent with its components.

Scalar Multiplication

Multiplying a vector by a scalar changes its magnitude but not its direction.

Step 1: Multiply each component of the vector by the scalar. For $𝑘\mathbf{𝑎}$, if 𝑘 = 3 and $\mathbf{𝑎} = \begin{pmatrix} 𝑎_1 \\ 𝑎_2 \end{pmatrix}$, then $3\mathbf{𝑎} = \begin{pmatrix} 3𝑎_1 \\ 3𝑎_2 \end{pmatrix}$.
Why: This stretches or shrinks the vector, making it longer or shorter.

Dot Product

The dot product of two vectors gives a scalar (a plain number) and shows how much one vector goes in the direction of another.

Step 1: Multiply corresponding components and add them up. For vectors $\mathbf{𝑎}$ and $\mathbf{𝑏}$, $\mathbf{𝑎} \cdot \mathbf{𝑏} = 𝑎_1𝑏_1 + 𝑎_2𝑏_2$.
Why: This measures the projection of one vector onto another, useful for finding angles.

Quick check

1. Add $\mathbf{𝑎} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{𝑏} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}$.
2. Find the scalar multiple $3\mathbf{𝑎}$ if $\mathbf{𝑎} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$.
3. Calculate the dot product of $\mathbf{𝑎} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{𝑏} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.

Answers:

1. $\begin{pmatrix} 6 \\ 4 \end{pmatrix}$
2. $\begin{pmatrix} 6 \\ -3 \end{pmatrix}$
3. $11$

Revision checklist

• Rushing Steps: Slow down when adding or subtracting components. Check each component separately.
• Forgetting Scalar Multiplication: Remember to multiply all components by the scalar, not just one.
• Overcomplicating Dot Product: Focus on multiplying and adding components accurately.
• Units and Directions: Always check if your final vector answer has correct units and directions.

Exam tip

Vectors questions in exams often test your ability to apply rather than just recall. Look for key words like "magnitude" or "direction" and relate them to what you've learned. Draw diagrams if needed to visualise the problem.

Worked examples

Question 1

Find the vector sum of $\mathbf{𝑢} = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}$ and $\mathbf{𝑣} = \begin{pmatrix} 1 \\ 4 \\ -3 \end{pmatrix}$.

Solution

Step 1: Add corresponding components:
$\mathbf{𝑢} + \mathbf{𝑣} = \begin{pmatrix} 3 + 1 \\ -2 + 4 \\ 5 - 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 2 \\ 2 \end{pmatrix}$.
Why: By adding each component, we ensure the new vector has the correct direction and magnitude.

Question 2

Calculate the dot product of $\mathbf{𝑎} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{𝑏} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$.

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Solution

Step 1: Multiply corresponding components and sum them up:
$\mathbf{𝑎} \cdot \mathbf{𝑏} = (2)(-1) + (3)(4) = -2 + 12 = 10$.
Why: This gives us how much $\mathbf{𝑎}$ goes in the direction of $\mathbf{𝑏}$, a useful scalar.

Quick summary

• Vectors have both direction and magnitude, similar to an arrow.
• Add and subtract vectors by combining components.
• Scalar multiplication changes the length of the vector.
• Dot product results in a scalar, useful for angles.
• Practice with diagrams to visualise problems.
• Slow down to avoid careless mistakes during exams.
• Recognize keywords in questions to apply correct methods.

FAQ

Why do I get stuck on vector questions?
Many students freeze because they try to memorise rather than understand the concept. Visualising vectors can help.

How do I know when to use the dot product?
Use it when the question involves angles or projections between vectors.

What if I make calculation errors often?
Check each step methodically and practice breaking down the components. Double-check your arithmetic.

Are vectors really important for exams?
Yes, they test your understanding of space and direction, which are crucial in many math problems.

Practise with free question sets

Work through exam-style questions with answers and step-by-step solutions:

• [35+ A Level H 2 Vectors Questions for 2026/2027 (Singapore MOE Syllabus) with Exam-Style Solutions](/questions/jc-h 2-math-vectors-questions)
• [Topic study hub](/learn/jc-h 2-math-vectors)

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Related Topics You Should Learn Next

• [JC H 2 Math: Vectors](https://tutorly.sg/learn/jc-h 2-math-vectors)
• [Vector Calculations for A Levels](https://tutorly.sg/blog/jc-h 2-math-vectors-guide)

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