Vector proofs: midpoints, ratios and collinear points

Vectors can prove that lines are parallel or that points lie on a straight line, without measuring anything.

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Routes and ratios

OA = a and OB = b, so AB = AO + OB = −a + b = b − a.
M is the midpoint of AB: OM = a + 12(b − a) = 12(a + b).
P divides AB in the ratio 2 : 1: AP = 23AB, so OP = a + 23(b − a) = 13a + 23b.

Parallel vectors

Two vectors are parallel if one is a number times the other.
2a + 6b = 2(a + 3b), so it is parallel to a + 3b and twice as long.
3a + kb is parallel to 2a + 4b when 32 = k4, so k = 6.

Collinear points

To prove A, B and C lie on one straight line:
1. Show AB and BC are parallel (one is a multiple of the other).
2. Say they share the point B.
3. Conclude A, B and C are collinear.
The magnitude of a column vector (x over y) is √(x2 + y2).
Worked example

AB = 2a + 6b and BC = a + 3b. Prove A, B and C are collinear.

  1. 2a + 6b = 2(a + 3b), so AB = 2BC.
  2. AB and BC are parallel.
  3. They share the point B.

Answer: A, B and C lie on a straight line

Key idea

Write each vector as a route through known vectors. Parallel means a multiple. Collinear means parallel and sharing a point.

Key words
Vector
A quantity with size and direction, written as a column vector or in bold, such as a.
Ratio
A comparison of two or more quantities, written a : b.

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. OA = a and OB = b. P is on AB with AP : PB = 2 : 1. Find OP.

    1. a + 23b
    2. 23a + 13b
    3. 13a + 23b
    4. 23(a + b)
    Show the answer

    Answer: 13a + 23b

    OP = a + 23(b − a) = 13a + 23b.

  2. OA = a and OB = b. M is the midpoint of AB. Find AM.

    1. 12(b − a)
    2. 12(a − b)
    3. 12(a + b)
    4. b − a
    Show the answer

    Answer: 12(b − a)

    AB = b − a, so AM = 12(b − a).

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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