IGCSE Maths vectors: column vectors, magnitude and vector geometry

What each tier expects

The key ideas

A column vector is written with the x movement on top and the y movement below. In plain text we will write it as (x, y). Add vectors by adding the tops and the bottoms; multiply by a number by multiplying both parts. So if a = (3, -2) and b = (-1, 4), then a + b = (2, 2) and 3b = (-3, 12).

The magnitude is the length of the arrow. By Pythagoras, |(x, y)| = sqrt(x^2 + y^2), so |(5, -3)| = sqrt(34) = 5.83 to 3 s.f.

In geometry questions you are told some vectors, such as OA = a and OB = b, and asked for others. Find a route along known vectors from the start to the end. Going against an arrow changes the sign: AO = -a. So AB = AO + OB = -a + b = b - a. For a point part way along a line, take a fraction of that line's vector: the midpoint M of AB gives OM = a + 1/2(b - a) = 1/2(a + b).

Vectors are parallel if one is a scalar multiple of the other: 4a + 2b is parallel to 2a + b because it is twice it. If PQ and QR are parallel and share the point Q, then P, Q and R lie on one straight line, which is what 'collinear' means.

Worked example 1: column vector arithmetic

1. a = (3, -2) and b = (-1, 4). Find 2a - 3b and |2a - 3b|.
2. 2a = (6, -4) and 3b = (-3, 12).
3. 2a - 3b = (6 - (-3), -4 - 12) = (9, -16).
4. |2a - 3b| = sqrt(9^2 + (-16)^2) = sqrt(81 + 256) = sqrt(337) = 18.4 to 3 s.f.

Worked example 2: a point dividing a line in a ratio

1. OA = a and OB = b. P lies on AB with AP : PB = 1 : 2. Find OP in terms of a and b.
2. AB = AO + OB = -a + b = b - a.
3. AP is 1/3 of AB, because the ratio 1 : 2 splits AB into 3 equal parts. So AP = 1/3(b - a).
4. OP = OA + AP = a + 1/3(b - a) = 2/3 a + 1/3 b, which can be written 1/3(2a + b).
5. Check with numbers: if a = (3, 0) and b = (0, 3), P should be one third of the way from A to B, at (2, 1). The formula gives 1/3(6, 3) = (2, 1). Correct.

Worked example 3: proving points are collinear

1. Using example 2, Q is the point with OQ = 4a + 2b. Show that O, P and Q are collinear.
2. OP = 1/3(2a + b), and OQ = 4a + 2b = 2(2a + b).
3. So OQ = 6 × OP, because 6 × 1/3(2a + b) = 2(2a + b).
4. OQ is a multiple of OP, so they are parallel, and they share the point O. Therefore O, P and Q are collinear. State both facts in the conclusion: parallel and a common point.

Common mistakes that cost marks

• Forgetting the sign when travelling against an arrow: AO = -a, not a.
• Writing AB = a - b instead of b - a. The rule is end minus start when both start from O.
• Using the wrong fraction for a ratio: AP : PB = 1 : 2 means AP is 1/3 of AB, not 1/2.
• Concluding 'collinear' from parallel vectors without mentioning the shared point.
• Squaring a negative component incorrectly in a magnitude: (-16)^2 is 256.
• Not simplifying the final expression, for example leaving a + 1/3(b - a) unexpanded when the question asks for the simplest form.

Exam technique and how a tutor helps

Vector geometry is usually one multi-part question near the end of the paper, worth around 5 to 8 marks, so it is a grade-deciding topic for students aiming at the top grades. Write the route before the algebra, for example 'OP = OA + AP', because that line shows the examiner your method. Mark every new vector you find on the diagram, with an arrow showing its direction.

Most 'show that' parts end with a sentence. Say exactly what you have shown, such as 'OQ = 6 OP so OQ is parallel to OP, and both pass through O, so O, P and Q are collinear.'

Students often find vectors abstract until they see that the letters are just arrows on a diagram. A tutor working one to one can draw the routes on a shared whiteboard as the student describes them, and then gradually remove that support. Lessons usually move from column-vector arithmetic to ratio problems to collinearity proofs, using past-paper questions from the student's own board at each stage.

Self-check: can you do these?

• Work out (2, 5) + (3, -1). (Answer: (5, 4))
• Work out 3 × (-2, 1). (Answer: (-6, 3))
• Find the magnitude of (6, 8). (Answer: 10)
• OA = a, OB = b and M is the midpoint of AB. Find OM. (Answer: 1/2(a + b))
• Is 6a - 9b parallel to 2a - 3b? Explain. (Answer: yes, it is 3 times it)