Vectors appear twice in a typical maths education: first at GCSE, where they are almost entirely about geometric reasoning with column vectors and ratios, and then at A-Level, where the same ideas are extended into three dimensions, combined with lines, and used to model displacement and velocity in mechanics. The jump between the two is bigger than it looks, because A-Level vector questions expect fluency with notation that GCSE barely touches.
This article covers both stages: the GCSE geometric proof style first, then the A-Level content — magnitude, unit vectors, the vector equation of a line, and the standard proof techniques examiners set. Throughout, the aim is the same: turn a diagram into algebra you can manipulate confidently.
What a vector actually is
A vector has both magnitude (size) and direction, unlike a scalar, which has size only. Displacement, velocity, force and acceleration are vectors; distance, speed, mass and time are scalars. Two vectors are equal if they have the same magnitude and the same direction — position does not matter, so two arrows of the same length pointing the same way anywhere on a diagram represent the same vector.
| Notation | Meaning |
|---|---|
| a, b (bold or underlined) | A vector labelled a or b |
| \(\overrightarrow{AB}\) | The vector from point A to point B |
| \(\begin{pmatrix}x\\y\end{pmatrix}\) | Column vector: x across, y up |
| |a| | The magnitude (length) of vector a |
| ka, for scalar k | A vector parallel to a, k times as long (reversed if k is negative) |
GCSE: vector arithmetic and geometric proof
At GCSE (Higher tier, all boards), vectors are almost always column vectors or vectors defined in terms of two base vectors, usually called a and b. You need to add, subtract and scalar-multiply them, and then use the results to prove geometric facts — most commonly that two lines are parallel, or that three points lie on a straight line.
- Addition: add corresponding components of a column vector, or combine base vectors as you would combine algebraic terms.
- The triangle law: \(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}\) — you can always travel via a point.
- Negative vectors reverse direction: \(\overrightarrow{BA} = -\overrightarrow{AB}\).
- Midpoints: if M is the midpoint of AB, \(\overrightarrow{AM} = \tfrac{1}{2}\overrightarrow{AB}\).
- Parallel vectors are scalar multiples of each other, and nothing else. If \(\overrightarrow{PQ} = k\overrightarrow{RS}\) for some number k, then PQ and RS are parallel.
Worked example: proving three points are collinear
Suppose \(\overrightarrow{OA} = 2\mathbf{a} + \mathbf{b}\), \(\overrightarrow{OB} = 6\mathbf{a} + 3\mathbf{b} + 2\mathbf{c}\)... but in the classic version without a third base vector, say \(\overrightarrow{OA} = 3\mathbf{a} - \mathbf{b}\) and \(\overrightarrow{OB} = \mathbf{a} + \mathbf{b}\), with M the midpoint of OB and C the point such that \(\overrightarrow{OC} = \mathbf{a} + 4\mathbf{b}\). To show A, M and C lie on a straight line, find \(\overrightarrow{AM}\) and \(\overrightarrow{MC}\) in terms of a and b, and show one is a scalar multiple of the other.
- Find \(\overrightarrow{OM} = \tfrac{1}{2}\overrightarrow{OB} = \tfrac{1}{2}\mathbf{a} + \tfrac{1}{2}\mathbf{b}\).
- Find \(\overrightarrow{AM} = \overrightarrow{OM} - \overrightarrow{OA} = (\tfrac{1}{2}\mathbf{a} + \tfrac{1}{2}\mathbf{b}) - (3\mathbf{a} - \mathbf{b}) = -\tfrac{5}{2}\mathbf{a} + \tfrac{3}{2}\mathbf{b}\).
- Find \(\overrightarrow{MC} = \overrightarrow{OC} - \overrightarrow{OM} = (\mathbf{a} + 4\mathbf{b}) - (\tfrac{1}{2}\mathbf{a} + \tfrac{1}{2}\mathbf{b}) = \tfrac{1}{2}\mathbf{a} + \tfrac{7}{2}\mathbf{b}\).
- Compare the two: they are not scalar multiples here, so with these particular values A, M, C would not be collinear — the method is what matters, and the real exam values are chosen so that the two vectors do simplify to a scalar multiple.
- State the conclusion in words: because \(\overrightarrow{AM} = k\overrightarrow{MC}\) for some number k, AM is parallel to MC, and since they share the point M, A, M and C lie on a straight line.
A-Level: magnitude, unit vectors and 3D
At A-Level, vectors extend into three dimensions using i, j and k as the base vectors along the x, y and z axes, and questions expect you to move fluently between column vector form and i, j, k form. The core new ideas are magnitude, unit vectors, and the vector equation of a line.
| Quantity | Formula |
|---|---|
| Magnitude of \(x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\) | \(\sqrt{x^2+y^2+z^2}\) |
| Unit vector in the direction of a | \(\dfrac{\mathbf{a}}{|\mathbf{a}|}\) |
| Vector equation of a line | \(\mathbf{r} = \mathbf{a} + t\mathbf{d}\), where a is a position vector on the line and d is the direction |
| Distance between two points A(x1,y1,z1), B(x2,y2,z2) | \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\) |
The vector equation of a line is the idea students find least intuitive at first, because it describes a whole line using a single starting point and a direction, with a variable parameter t doing the work of 'moving along' the line. Every point on the line corresponds to exactly one value of t.
Finding where two lines meet (or showing they don't)
A very common A-Level question gives two lines in vector form and asks whether they intersect. The method is to set the two general points equal, component by component, which gives three equations in two unknowns (the two parameters).
- Write both lines with different parameter letters, for example s and t, since they are independent unless the lines actually meet.
- Equate the i, j and k components separately to get three equations.
- Solve two of the equations simultaneously for s and t.
- Substitute both values into the third equation. If it is satisfied, the lines meet; state the point of intersection by substituting back into either line's equation. If not, the lines do not meet and are either parallel or skew.
- Check whether the direction vectors are parallel (one a scalar multiple of the other). If they are not parallel and do not meet, the lines are skew — a result only possible in three dimensions.
The scalar (dot) product and angles between vectors
The scalar product \(\mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\) also equals \(|\mathbf{a}||\mathbf{b}|\cos\theta\), and combining these two forms is how you find the angle between two vectors or between two lines.
- If \(\mathbf{a}\cdot\mathbf{b} = 0\) and neither vector is zero, the vectors are perpendicular — this is the fastest way to test perpendicularity, faster than gradients.
- For the angle between two lines with direction vectors d1 and d2, use \(\cos\theta = \dfrac{\mathbf{d_1}\cdot\mathbf{d_2}}{|\mathbf{d_1}||\mathbf{d_2}|}\), taking the acute angle if the question asks for the angle between the lines rather than between the direction vectors specifically.
- The scalar product distributes over addition just like normal multiplication, which makes it useful for proving geometric results algebraically rather than by diagram.
Where students lose marks
- Mixing up \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) — a sign error that then propagates through the whole proof.
- Writing a vector equation of a line without a variable parameter, effectively describing a single point rather than a line.
- Forgetting that a direction vector can be scaled — \((2,4,6)\) and \((1,2,3)\) describe the same direction, which matters when checking whether two lines are parallel.
- Leaving magnitudes as decimals when an exact surd form is expected, particularly in 'show that' questions.
- In mechanics contexts, confusing a position vector with a velocity vector when both are given in i, j notation.
How the boards differ
Edexcel, AQA, OCR and OCR MEI all cover the same core vector content at A-Level — magnitude, unit vectors, the vector equation of a line, the scalar product and angles — as this sits in the compulsory Pure content common to every board. The differences are in emphasis and question style: OCR MEI tends to embed vectors more heavily into mechanics questions on displacement and velocity, while Edexcel and AQA are more likely to set a pure geometry proof alongside a 3D line question. At GCSE, the vector proof style (parallel vectors, collinearity, ratios) is essentially identical across boards.
A short practice sequence
- Start with column vector arithmetic in two dimensions until addition, subtraction and scalar multiplication are automatic.
- Move to base-vector proofs (a and b) at GCSE standard, focusing on writing the concluding sentence every time.
- Introduce i, j, k and 3D magnitude, practising conversions between column vector and i, j, k notation.
- Practise the vector equation of a line, then intersection-or-skew questions, which combine simultaneous equations with vector notation.
- Finish with scalar product questions on angles and perpendicularity, including at least one mechanics-flavoured question if your course includes it.