Vector geometry

• Represent vectors by directed line segments: an arrow from start to finish.
• A position vector starts at the origin O: ⟶OA = a. Then ⟶AB = b − a.
• Express any vector in terms of two given vectors such as a and b by following a route.
• Use vectors to prove lines parallel, prove three points collinear, and solve problems with ratios and similar triangles.

An arrow carries the vector

A vector can be drawn as a directed line segment: a line with an arrow showing which way it goes. Any segment with the same length and direction represents the same vector. In a parallelogram OABC with ⟶OA = a and ⟶OC = c, the opposite sides are equal vectors:

Vectors that start at the origin

The position vector of a point A is ⟶OA, the vector from the origin to A. If A is (2, 5), its position vector is 25. With position vectors a and b:

Follow a route you know

Most questions give two vectors, often a and b, along two sides of a shape. To find any other vector, write a route from its start to its end using only sides you know, then simplify. A ratio on a line tells you the fraction of that line: AN : NB = 3 : 1 means AN is ¾ of AB.

Scalar multiples

Two vectors are parallel when one is a number times the other. Three points are collinear (on one straight line) when two of the vectors joining them are parallel and share a point. Always finish with a sentence that says what you have shown.