A vector describes a movement with both a size (magnitude) and a direction. Unlike an ordinary number, a vector tells you not just how far to travel but which way. At GCSE, vectors are written as column vectors and used to describe paths between points in geometric diagrams.
What is a vector and how is it written?
A vector has two components: a horizontal movement and a vertical movement. In GCSE maths the column vector notation places the horizontal component on top and the vertical below, separated by a fraction line or in a bracket. A positive top number means move right; negative means move left. A positive bottom number means move up; negative means move down.
Vectors are labelled in bold in printed text (e.g. a) or underlined in handwriting. The vector from point A to point B is written with an arrow above: AB with an arrow. A vector has magnitude (length) and direction, but no fixed position — two arrows of the same length pointing the same way represent the same vector even if they are in different places on the diagram.
How do you add two vectors?
To add two vectors, add the horizontal components and add the vertical components separately.
Worked example: If a has components (4, 1) and b has components (−1, 3), find a + b.
a + b = (4 + (−1), 1 + 3) = (3, 4)
Geometrically, adding vectors means placing them tip-to-tail: travel along a then travel along b — the result is the single vector from start to finish.
| Operation | Horizontal component | Vertical component |
|---|---|---|
| a | 4 | 1 |
| b | −1 | 3 |
| a + b | 4 + (−1) = 3 | 1 + 3 = 4 |
How do you subtract vectors?
Subtracting b from a is the same as adding the negative of b. The negative of a vector reverses its direction: if b = (−1, 3), then −b = (1, −3).
Worked example: Find a − b where a = (4, 1) and b = (−1, 3).
a − b = (4 − (−1), 1 − 3) = (5, −2)
How do you multiply a vector by a scalar?
A scalar is an ordinary number. Multiplying a vector by a scalar scales both components by that number.
Worked example: If a = (3, −2), find 4a.
4a = (4 × 3, 4 × (−2)) = (12, −8)
The direction stays the same; the magnitude is multiplied by 4. If the scalar is negative, the direction reverses: −2a = (−6, 4).
How do you find the magnitude of a vector?
The magnitude of vector v = (x, y) is found using Pythagoras: |v| = √(x² + y²).
Worked example: Find the magnitude of a = (3, 4).
|a| = √(3² + 4²) = √(9 + 16) = √25 = 5 units
How do you navigate vector paths in geometry questions?
Geometry questions define two or more vectors using simple letters, then ask you to express the path between two points as a combination of those vectors.
Key rule: To travel from A to C via B: →AC = →AB + →BC.
Worked example: In a diagram, →OA = a and →OB = b. M is the midpoint of AB. Find →OM in terms of a and b.
- →AB = →AO + →OB = −a + b = b − a.
- →AM = ½ × →AB = ½(b − a).
- →OM = →OA + →AM = a + ½(b − a) = a + ½b − ½a = ½a + ½b.
So →OM = ½(a + b). The vector to the midpoint is the average of the two position vectors — a useful pattern to remember.
What mistakes do students commonly make?
- Getting the direction of →AB wrong. →AB goes from A to B; →BA goes from B to A, so →BA = −→AB. Always trace the arrow direction carefully.
- Adding magnitudes instead of components. Two vectors (3, 0) and (0, 4) have magnitudes 3 and 4, but their sum is (3, 4) with magnitude 5 — not 3 + 4 = 7.
- Forgetting to negate when travelling backwards. If →OA = a, travelling from A to O gives −a.
Frequently asked questions
What is the difference between a vector and a scalar?
A scalar has magnitude only — for example, speed (50 mph) or temperature (20 °C). A vector has both magnitude and direction — for example, velocity (50 mph north) or displacement. In GCSE maths you manipulate vectors component by component, not just as single numbers.
Are vectors only on the Higher GCSE paper?
Yes — vector geometry appears on the Higher tier only at GCSE. On Edexcel, AQA, and OCR Higher papers, vector questions typically ask students to express paths in terms of given vectors, identify midpoints, and sometimes prove that three points are collinear. They are worth 4–6 marks and reward clear, methodical working.
What does it mean for two vectors to be parallel?
Two vectors are parallel if one is a scalar multiple of the other. For example, (6, −4) and (−3, 2) are parallel because (6, −4) = −2 × (−3, 2). In geometry proofs, showing that two path vectors are scalar multiples of each other proves those line segments are parallel.
How do you show three points are collinear using vectors?
Find the vector between two pairs of the points. If →AB = k × →AC for some scalar k, then A, B, and C lie on the same straight line. You must also state that they share a common point to confirm they are collinear rather than merely on parallel lines.
For Socratic GCSE geometry and vectors practice, see aitutors.me.