"A length and an angle" describes a vector, but it's awkward to compute with. There's a tidier way. On the coordinate plane, any arrow from the origin is fixed by where its head lands. So we can describe the whole vector with just two numbers: how far it goes across and how far it goes up.

These two numbers are the vector's components. We stack them in a column:

For example \begin{bmatrix} 3 \\ 2 \end{bmatrix} means "go 3 right, then 2 up." The arrow and the pair of numbers are two faces of the same thing — that's the bridge linear algebra is built on.

Reading off the components

Steer the head of the vector below. The dashed legs show its two components — the across step v_x and the up step v_y — and the column updates to match. Every arrow you can draw has exactly one such pair, and every pair draws exactly one arrow.

Points, displacements, and the zero vector

A vector \begin{bmatrix} 3 \\ 2 \end{bmatrix} can name a position (the point at (3, 2)) or a displacement (the move "3 right, 2 up" from wherever you are). Same numbers, two readings — and both are useful.

The special vector \vec{0} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} is the zero vector: no length, no direction, an arrow that goes nowhere. And nothing forces us to stop at two components — a vector in three dimensions has three, and a data point with 100 features is a vector with 100 components. The picture stops at 3D; the arithmetic never does.