Question

How do you obtain a unit vector for a nonzero vector in a given direction?

Answer 1

A vector of norm 1 is called a unit vector. Such vectors are useful for specifying a direction when length is not relevant to the problem at hand. You can obtain a unit vector in a desired direction by choosing any nonzero vector $\mathbf{v}$ in that direction and multiplying $\mathbf{v}$ by the reciprocal of its length. For example, if $\mathbf{v}$ is a vector of length 2 in $R^{2}$ or $R^{3}$, then $\frac{1}{2} \mathbf{v}$ is a unit vector in the same direction as $\mathbf{v}$. More generally, if $\mathbf{v}$ is any nonzero vector in $R^{n}$, then $$ \mathbf{u}=\frac{1}{\|\mathbf{v}\|} \mathbf{v} $$