a unit vector

Sometimes you care only about which way something points, not how strong it is — the compass heading, not the speed. A unit vector is exactly that: a vector of length one, carrying pure direction and nothing else. Think of it as a tiny standardised arrow you can later scale up to any size you need.

Any nonzero vector v can be turned into a unit vector pointing the same way by dividing it by its own length. This is called normalising. Write the result v-hat = v / |v|; then |v-hat| = 1 and v-hat points exactly along v. For example, if v = (3, 0, 4) then |v| = sqrt(9 + 16) = 5, so v-hat = (3/5, 0, 4/5). The three standard basis vectors i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) are themselves unit vectors, one along each axis. Conversely, every vector can be rebuilt as 'its length times its direction': v = |v| (v-hat), which neatly separates the 'how much' from the 'which way'.

Unit vectors are the natural way to specify a direction in lines, planes, and physics. The direction of a line in space, the outward normal of a surface, the line of action of a force — all are most cleanly given as unit vectors, so that the magnitude can be supplied separately. The one rule you cannot break: the zero vector (0, 0, 0) has length zero and points nowhere, so it can never be normalised. Dividing by |0| = 0 is undefined, which is the algebra's honest way of saying 'a point with no direction has no direction to report'.