Question

What is the norm of a vector?

Answer 1

The norm or length of a vector $u$ in $\mathbf{R}^{n}$, denoted by $\|u\|$, is defined to be the nonnegative square root of $u \cdot u .$ In particular, if $u=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$, then
$$
\|u\|=\sqrt{u \cdot u}=\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}
$$
That is, $\|u\|$ is the square root of the sum of the squares of the components of $u$. Thus, $\|u\| \geq 0$, and $\|u\|=0$ if and only if $u=0$.

A vector $u$ is called a unit vector if $\|u\|=1$ or, equivalently, if $u \cdot u=1$. For any nonzero vector $v$ in $\mathbf{R}^{n}$, the vector
$$
\hat{v}=\frac{1}{\|v\|} v=\frac{v}{\|v\|}
$$
is the unique unit vector in the same direction as $v$. The process of finding $\hat{v}$ from $v$ is called normalizing $v$.