Which of the following is a vector?
A vector has direction and magnitude.
(3, 2) represents a movement in x and y directions.
Vectors
Vectors describe movement using direction and magnitude. They are used in geometry and link closely to coordinates and transformations.
Overview
A vector tells you both size and direction.
In GCSE Maths, vectors are often used to describe movement from one point to another.
\( \begin{pmatrix} 3 \\ 2 \end{pmatrix} \)
This vector means move 3 units to the right and 2 units up.
Unlike normal lengths, vectors include direction as well as distance.
What you should understand after this topic
- Understand what a vector represents
- Write vectors in column form
- Describe movement using vectors
- Add and subtract vectors
- Understand how vectors appear in geometry and proof questions
Key Definitions
Vector
A quantity with both size and direction.
Scalar
A quantity with size only, such as length or mass.
Magnitude
The size or length of a vector.
Direction
The way the vector points.
Column Vector
A vector written vertically in brackets.
Directed Line Segment
A line showing both distance and direction from one point to another.
Key Rules
Top number = horizontal
Positive means right, negative means left.
Bottom number = vertical
Positive means up, negative means down.
Add matching parts
Add the top numbers together and the bottom numbers together.
Subtract matching parts
Subtract top from top and bottom from bottom.
Quick Movement Guide
\(\begin{pmatrix} 4 \\ 0 \end{pmatrix}\)
4 right
\(\begin{pmatrix} -2 \\ 0 \end{pmatrix}\)
2 left
\(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\)
5 up
\(\begin{pmatrix} 0 \\ -3 \end{pmatrix}\)
3 down
How to Solve
Step 1: Read a vector as movement
A column vector shows horizontal and vertical movement.
\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
Top = horizontal movement.
Bottom = vertical movement.
This means 5 right and 2 up.
Step 2: Negative components
\(\begin{pmatrix} -3 \\ -4 \end{pmatrix}\)
Negative top → left.
Negative bottom → down.
Step 3: Add and subtract
\(\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}\)
Add top with top, bottom with bottom.
Step 4: Multiply by a scalar
\(3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}\)
Step 5: Vector between two points
Use B − A.
\(A(2,1),\ B(7,4)\)
Horizontal: \(7 - 2 = 5\)
Vertical: \(4 - 1 = 3\)
\(\begin{pmatrix}5 \\ 3\end{pmatrix}\)
Step 6: Exam method
See transformations for translation vectors.
- Read horizontal (top).
- Read vertical (bottom).
- Use negatives for left/down.
- Add/subtract components.
- Use B − A for direction.
Example Questions
Edexcel
Exam-style questions focusing on reading column vectors as movements.
Edexcel
The vector \( \begin{pmatrix} 2 \\ 5 \end{pmatrix} \) is shown on the grid.
Describe the movement.
Edexcel
The vector \( \begin{pmatrix} -4 \\ 1 \end{pmatrix} \) is shown on the grid.
Describe the movement.
AQA
Exam-style questions focusing on adding and subtracting column vectors.
AQA
Work out \( \begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 4 \end{pmatrix} \).
AQA
Work out \( \begin{pmatrix} 6 \\ 1 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} \).
OCR
Exam-style questions focusing on vectors between points and describing translations.
OCR
Point A has coordinates (1, 2) and point B has coordinates (5, 7).
Find the vector from A to B.
OCR
Point P is moved by vector \( \begin{pmatrix} -3 \\ 4 \end{pmatrix} \).
Describe the movement.
OCR
A shape is translated by \( \begin{pmatrix} 4 \\ -2 \end{pmatrix} \).
Describe the translation.
Exam Checklist
Step 1
Check which part is horizontal and which is vertical.
Step 2
Watch the signs carefully.
Step 3
Add or subtract the matching numbers only.
Step 4
If working between points, subtract coordinates in the correct order.
Most common exam mistakes
Wrong direction
Positive and negative movements mixed up.
Wrong order
Top and bottom parts confused.
Wrong subtraction
Subtracting the coordinates in reverse.
Coordinate confusion
Treating a vector like a plotted point rather than a movement.
Common Mistakes
These are common mistakes students make when working with vectors in GCSE Maths.
Mixing up horizontal and vertical movement
Incorrect
A student swaps the x and y components of a vector.
Correct
Vectors are written as \(\begin{pmatrix} x \\ y \end{pmatrix}\), where x is horizontal movement and y is vertical movement.
Misunderstanding negative values
Incorrect
A student ignores the direction of negative components.
Correct
Negative values show direction: negative x is left, negative y is down.
Subtracting in the wrong order
Incorrect
A student reverses the subtraction when finding a vector.
Correct
To find the vector from A to B, calculate \(B - A\), not \(A - B\).
Treating vectors like coordinates
Incorrect
A student reads a vector as a point on a grid.
Correct
A vector represents movement, not position. It shows how far and in what direction to move.
Arithmetic errors in calculations
Incorrect
A student makes mistakes when adding or subtracting vectors.
Correct
Add and subtract vectors component-wise and check calculations carefully.
Try It Yourself
Practise solving problems involving vectors.
Foundation
Foundation Practice
Understand vector notation, directions and simple vector addition/subtraction.
Add the vectors (2, 3) + (4, 1).
Add x-values and y-values separately.
(2+4, 3+1) = (6, 4).
What is the vector from A to B?
Count across then up/down.
From A to B is 2 right and 2 up/down → (2, -2).
Find (5, 7) − (2, 3).
Subtract coordinates.
(5−2, 7−3) = (3, 4).
What is 2 × (3, 4)?
Multiply both components.
2 × (3, 4) = (6, 8).
Find the vector from (1, 2) to (5, 6).
Subtract starting point from ending point.
(5−1, 6−2) = (4, 4).
Which vector is opposite to (4, -3)?
Reverse both signs.
Opposite vector is (-4, 3).
Find (2, 5) + (−3, 4).
Add x and y separately.
(2−3, 5+4) = (−1, 9).
What does the vector (0, 5) represent?
x = 0 means no horizontal movement.
(0, 5) means move 5 units up.
Find the midpoint of (2, 2) and (6, 6).
Average x and y values.
(2+6)/2 = 4 and (2+6)/2 = 4.
Higher
Higher Practice
Solve vector proofs, algebraic vectors and geometric vector problems.
Find the vector AB if A = (2, 3) and B = (7, 11).
B − A.
(7−2, 11−3) = (5, 8).
If a = (2, 1) and b = (4, 3), what is a + b?
Add components.
(2+4, 1+3) = (6, 4).
If a = (3, 2) and b = (1, 4), find 2a − b.
Multiply then subtract.
2a = (6,4), so (6,4) − (1,4) = (5,0).
Points A, B and C are collinear. AB = (2, 3) and BC = (4, 6). What can you say?
BC is a multiple of AB.
(4,6) = 2 × (2,3), so they are parallel and same direction.
Find the midpoint of A(1, 5) and B(7, 9).
Average coordinates.
(1+7)/2 = 4 and (5+9)/2 = 7.
Which expression represents the midpoint of vectors a and b?
Think average.
Midpoint = (a + b)/2.
If a = (4, 6), find a vector in the same direction but half the length.
Divide by 2.
(4,6) ÷ 2 = (2,3).
If AB = (3, 5), what is BA?
Reverse direction.
BA is the opposite vector: (-3, -5).
Find the vector from (−2, 4) to (3, −1).
Subtract coordinates.
(3 − (−2), −1 − 4) = (5, −5).
Which condition proves two vectors are equal?
Both size and direction matter.
Vectors are equal if they have the same direction and magnitude.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is a vector?
A quantity with direction and magnitude.
How are vectors written?
As column vectors or with arrows.
What does vector addition mean?
Combining movements.