Multiplying a vector by an ordinary number — a scalar — stretches or shrinks it without changing the line it lies along. Double a vector and it gets twice as long, same direction. Halve it and it shrinks. Multiply by a negative number and it flips to point the other way.

In components, multiply every entry by the scalar:

So 3\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 6 \\ 3 \end{bmatrix}, and -1\begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} -2 \\ -1 \end{bmatrix}.

Stretch, shrink, flip

Drag the scalar c below. The faint arrow is the original \vec{v}; the bold one is c\,\vec{v}. Watch how it always stays on the same line through the origin — scaling can lengthen, shorten or reverse a vector, but never tilt it.

The words for it

All the multiples of a single non-zero vector — every c\,\vec{v} as c ranges over all numbers — trace out an entire line through the origin. Scaling by 0 gives the zero vector; scaling by 1 leaves the vector unchanged. Two vectors that are scalar multiples of each other are called parallel (or collinear) — a fact we'll lean on when we meet linear independence.