If $v\not=0$, then $v/\|v\|$ has norm 1

Question

The question is:

Show that if $\vec v$ is a non-zero vector in $\mathbb R^n$ then $\left( \dfrac{1}{||\vec v||} \right ) \vec v$ has norm $1$.

I assume that $\vec v=(v_1,v_2,v_3,...,v_n)$ , where $n= 1,2,...$

$$ ||\vec v||= \sqrt{v_1^2+v_2^2+...v_n^2} $$

but I try to find out the track to solve $\left( \dfrac{1}{||\vec v||} \right ) \vec v$ to get norm $1$ but unfortunately I couldn't.

Please help me..

Answer 1

Just compute the norm of the vector ${1\over \|v\|}v={v\over \|v\|}$: $$\left\|{v\over \|v\|}\right\|={1\over \|v\|}\|v\|=1.$$

On the first equality, we have used the (absolute) homogeneous property norms.