Directed line segment notation revision notes

Edexcel International GCSE Maths (4MA1), chapter 27: Vectors, specification point 5.1b. Free lesson.

Watch the video lesson · IGCSE Maths past papers

A directed line segment names a vector by its start point and finish point. The order matters because reversing the journey gives the negative vector, a rule you need when reading diagrams and writing vector answers.

How to read a directed line segment

A directed line segment is a vector written with two letters and an arrow above them, such as →AB. The arrow runs from the first letter to the second, so you start at A and finish at B. The order is part of the vector's meaning. →AB starts at A and finishes at B. Reversing the journey gives the negative: →BA = −→AB.

The two ways to write a vector

Vectors may be named by a single bold letter or by the two end points with an arrow above them. A bold letter is given in the question, while the two-letter notation comes from the start and finish points in the diagram. Both notations can appear in the same question. In print, a bold a and →AB are both ways to name a vector. When writing by hand, underline a letter to show that it is bold. Reversing either notation changes the sign: the reverse of a is −a, and →BA = −→AB.

Find a vector by travelling through a known point

If a diagram does not show a direct arrow between the two points you need, break the journey into shorter parts through a point you know. Add the vectors for each part. For the triangle with →OA = a and →OB = b, the route from A to B goes back to O, then out to B. →AO = −a and →BO = −b, because each journey reverses a vector that starts at O. To travel from A to B, use →AO + →OB = −a + b. The journey from B to A reverses every part, so →BA = a − b.

Check the order of the letters

The first letter always marks the start. A route and its answer can be correct, but writing the letters in the wrong order gives the opposite vector. For example, →CO is the journey from C to O, so it is the reverse of →OC. →CO = −→OC. →AB and →BA lie along the same line, but point in opposite directions, so they are not the same vector. Before writing your answer, read the two letters back as a journey.

Apply the route method

For →OP = p and →OQ = q, find →QP by travelling from Q back to O, then from O to P. The first part is −q and the second part is p, so the result is p − q.