Sometimes you only care about which way a vector points, not how long it is. A unit vector is a vector of length exactly 1 — pure direction, no magnitude to distract you. We mark it with a hat: \hat{v}.

To turn any non-zero vector into a unit vector pointing the same way, divide it by its own length. This is called normalizing:

Dividing by the length is just scaling by 1/\lVert\vec{v}\rVert, so the direction is untouched and the new length comes out to 1.

Standing on the unit circle

Steer \vec{v} below. The short bold arrow is its normalized version \hat{v} — same heading, but its tip always lands on the unit circle (radius 1), no matter how long \vec{v} gets.

The standard unit vectors

Two unit vectors are so useful they get their own names. \hat{\imath} points one step along x, and \hat{\jmath} points one step along y:

Every vector is a linear combination of these two: \begin{bmatrix} 3 \\ 2 \end{bmatrix} = 3\hat{\imath} + 2\hat{\jmath}. They are the standard yardsticks the whole coordinate grid is measured against — and the first hint that a couple of well-chosen vectors can describe everything.