Answer
(a) $\|u\|=2\sqrt{2}, \quad \|v\|={\sqrt{5}}$.
(b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\frac{\sqrt{5}}{5}\left(\sqrt{2},-1,-\sqrt{2}\right)$.
(c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{\sqrt{2}}{4}\left(-1,\sqrt{3},2\right)$.
(d) ${u} \cdot {v} =-3\sqrt{2}-\sqrt{3}$.
(e)${u} \cdot {u}=8$.
(f) ${v} \cdot {v}=5$.
Work Step by Step
(a) $\|u\|=2\sqrt{2}, \quad \|v\|={\sqrt{5}}$.
(b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\frac{\sqrt{5}}{5}\left(\sqrt{2},-1,-\sqrt{2}\right)$.
(c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{\sqrt{2}}{4}\left(-1,\sqrt{3},2\right)$.
(d) ${u} \cdot {v} =-3\sqrt{2}-\sqrt{3}$.
(e)${u} \cdot {u}=8$.
(f) ${v} \cdot {v}=5$.
