Which Way and How Far? An Introduction to Vectors

Imagine you are giving directions to a friend standing in the middle of a park. You say: 'Walk 5 metres.' Your friend looks at you and asks: 'Which way?' You realise your instruction is incomplete — the distance alone is not enough. Your friend also needs a direction. This small problem is at the heart of one of maths most useful ideas: the vector. Before we dive in, try the warm-up question below.

An ordinary number that only measures size is called a scalar. The number 5 metres, 30 seconds, or £12 are all scalars — they tell you 'how much' but not 'which way'. A vector is different. A vector carries two pieces of information at once: its magnitude (size) and its direction. Saying 'move 5 metres to the right' is a vector idea. On a grid, we describe vectors using column vector notation. A column vector is written as two numbers stacked vertically inside brackets: ⎛ x ⎞ ⎝ y ⎠ The top number, x, tells you how far to move horizontally — positive means right, negative means left. The bottom number, y, tells you how far to move vertically — positive means up, negative means down. So the column vector (3, 2) means: move 3 units right and 2 units up. We often label a vector with a bold lowercase letter or an underlined letter, for example: a or a⃗. When we write it by hand we underline it: a. Adding Vectors When two vectors are added, each component is added separately: ⎛ a ⎞ ⎛ c ⎞ ⎛ a + c ⎞ ⎝ b ⎠ + ⎝ d ⎠ = ⎝ b + d ⎠ This is exactly like making two journeys one after the other — first vector a moves you somewhere, then vector b moves you on from there. The result is called the resultant vector.