Learn8300Papers 1–3 · Higher tierGeometry and measuresVector geometry

Geometry and measures · Subtopic 17 of 17

Vector Geometry

To find a vector between two points, follow any route along known vectors, adding as you go and subtracting when you travel backwards. Two vectors are parallel if one is a multiple of the other; points are collinear if they are joined by parallel vectors sharing a point.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HAQA 8300 Papers 1–3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Write vectors such as $\overrightarrow{AB}$ in terms of $\mathbf a$ and $\mathbf b$ by following a path.
Use midpoints and ratios on a line segment.
Show two vectors are parallel.
Show three points are collinear.

Pause and think

To get from $A$ to $B$ , you can walk $A\to O\to B$ . If $\overrightarrow{OA}=\mathbf a$ , what is the vector for walking from $A$ back to $O$ ?

Section 2 of 14

Core idea

Start hereProving lines are parallel without measuring

Draw a triangle, join the midpoints of two sides, and that line is always parallel to the third side, exactly half as long. Vectors prove it in two lines of algebra, no protractor needed.

The big idea

Route through known vectors: $\overrightarrow{AB}=\mathbf b-\mathbf a$ . A multiple ⇒ parallel; parallel and sharing a point ⇒ collinear.

1. Routes and midpoints

Midpoints in a triangle: MN is parallel to AB and half as long

\overrightarrow{AB}=\mathbf b-\mathbf a

and

\overrightarrow{MN}=\tfrac12\mathbf b-\tfrac12\mathbf a

, which is

\tfrac12\overrightarrow{AB}

: parallel and half the length.

Worked exampleMidpoint

$\overrightarrow{OA}=\mathbf a$ , $\overrightarrow{OB}=\mathbf b$ , $M$ midpoint of $AB$ .

Goal: $\overrightarrow{OM}$ .
Strategy: O → A → halfway along AB.

$\overrightarrow{AB}=-\mathbf a+\mathbf b$
A → O → B.
$\overrightarrow{OM}=\mathbf a+\tfrac12(\mathbf b-\mathbf a)=\tfrac12(\mathbf a+\mathbf b)$
Half of AB.

Answer

\frac12(\mathbf a+\mathbf b)

✓ CheckSymmetric in

\mathbf a

and

\mathbf b

, as a midpoint should be ✓.

Worked exampleRatio 1 : 3

$P$ divides $AB$ in ratio $1:3$ .

Goal: $\overrightarrow{OP}$ .
Strategy: $P$ is $\frac14$ of the way.

$\overrightarrow{AP}=\tfrac14(\mathbf b-\mathbf a)$
1 of 4 parts.
$\overrightarrow{OP}=\tfrac34\mathbf a+\tfrac14\mathbf b$
Add $\mathbf a$ .

Answer

\frac34\mathbf a+\frac14\mathbf b

✓ CheckNearer

A

, so more

\mathbf a

✓.

Worked exampleThe midpoint theorem

$M$ , $N$ midpoints of $OA$ , $OB$ . Show $MN\parallel AB$ .

Goal: Proof.
Strategy: Express both.

$\overrightarrow{MN}=-\tfrac12\mathbf a+\tfrac12\mathbf b=\tfrac12(\mathbf b-\mathbf a)$
M → O → N.
$=\tfrac12\overrightarrow{AB}$
A multiple.

AnswerParallel, and half the length.

✓ CheckMatches the hook ✓.

Pause and think

$\overrightarrow{PQ}=2\mathbf a-\mathbf b$ , $\overrightarrow{QR}=6\mathbf a-3\mathbf b$ . Conclusion?

Summary

Route through known vectors.
Multiple ⇒ parallel; plus a shared point ⇒ collinear.

Collinear: On one straight line.

HistoryGeometry without pictures. Vector methods let you prove geometric facts by algebra alone. In the 1800s Hermann Grassmann and William Rowan Hamilton developed the algebra; Josiah Willard Gibbs then shaped it into the vector notation taught today, partly to help his physics students.

Fun factThe centre of a triangle. Join each corner of a triangle to the midpoint of the opposite side. All three lines meet at one point, two-thirds of the way along each. Vectors prove it in a few lines: that point is

\tfrac13(\mathbf a+\mathbf b+\mathbf c)

, the triangle's balance point.

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