Vector Geometry Proofs

Grade 8

Three things to prove

Vector proofs at GCSE usually ask you to prove one of three things: that two lines are parallel, that a point is a midpoint, or that three points are collinear (lie on the same straight line).

Parallel: show one vector is a scalar multiple of the other (same direction). Collinear: show the vector from A to B and the vector from A to C are scalar multiples, and they share point A.

Setting up the path

Express every vector you need in terms of the given base vectors (a and b). Travel along known paths: go with the arrow → add, go against → subtract (use the negative). Use midpoint or ratio information to find intermediate points.

Write every vector path clearly, step by step. The algebra often simplifies nicely — if it looks messy, check you have the direction of each leg correct.

Worked example

EXAMPLE

OA = a, OB = b. M is the midpoint of AB. Show that OM = ½(a + b).

Vector AB = AO + OB = −a + b = b − a. M is the midpoint of AB, so AM = ½(b − a). Vector OM = OA + AM = a + ½(b − a) = a + ½b − ½a = ½a + ½b = ½(a + b) ✓.

Try this one

Past paper - 2023

OABC is a parallelogram.

\(\overrightarrow{OA}=a\) and \(\overrightarrow{OC}=c\)

The point N lies on line AB such that AN: NB = 3: 5.

(a) Find the following vectors in terms of a and c. Give your answers in their simplest form.

(i) \(\overrightarrow{OB}\)=------------[1]

(ii) \(\overrightarrow{ON}=\)----------- [2]

(b) Line CN is extended to reach point P, such that \(\overrightarrow{CP}=\frac85\overrightarrow{CN}\)

Show, using vectors, that OAP is a straight line. [4]

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