IGCSE Maths / Vectors

What is a vector revision notes

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

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A vector is a quantity with both magnitude (size) and direction. These two features tell you how vectors differ from scalars and when two vectors are equal.

What a vector is

A vector is a quantity with both a size, called its magnitude, and a direction. Both parts are needed for a complete definition. Saying only that a vector is an arrow or a movement does not give the mark. A scalar is a quantity with size only, such as 5 km of walking. A vector includes direction, such as 5 km due north. A scalar is written as a number. A vector can be shown in bold type or with an arrow over two letters, such as →AB.

How to find a vector's length and direction

For a = (3, 4), the components show 3 units across and 4 units up. These make the two shorter sides of a right-angled triangle. The vector is the sloping side, so use Pythagoras to find its length. Then use tangent to find the angle it makes with the horizontal. The length and direction are both needed to describe the vector fully.

When two vectors are equal

Two vectors are equal only when their magnitude and direction both match. Their starting positions do not matter, because a vector describes a move, not a location. You can slide an arrow across the page without changing the vector. Arrows of the same length are not necessarily equal if they point in different directions. Reversing a vector changes it to its negative: the length stays the same, but the direction is opposite.

How to compare two vectors

Check both parts of each vector: first its length, then the direction it points. Matching lengths alone are not enough. For p, which is 8 cm north, and q, which is 8 cm east, the lengths match but the directions do not. Therefore p ≠ q.

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