unit vector

A vector \mathbf u with \lVert\mathbf u\rVert = 1 is a unit vector. Any nonzero vector \mathbf v can be turned into a unit vector pointing the same way by normalizing it: \hat{\mathbf v} = \mathbf v / \lVert\mathbf v\rVert, since scaling by a positive c scales the length by that same factor, so dividing by \lVert\mathbf v\rVert always produces length exactly 1.