Proof that $R_u(v)=v-2\langle v,u\rangle u$ is an isometry on $\mathbb R^n$

Question

Fix a unit vector $u$ in $\mathbb R^n$. Define $R_u(v)=v-2\langle v,u\rangle u$ . Show that $R_u$ is an isometry .

And then this picture is given as help .

Please help , I cannot understand anything here .

Answer 1

Hint : show that $R_u$ preserves scalar products, i.e. $\langle R_u(v),R_u(v')\rangle =\langle v,v'\rangle$ for any $v,v'$. This is an algebraic computation.