Answer
(a) $\|u\|=\sqrt{3}, \quad \|v\|=2$.
(b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\left( -\frac{1}{2}, \frac{\sqrt{2}}{2},\frac{1}{2} \right)$.
(c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{\sqrt{3}}{3}\left(0, 1, \sqrt{2}\right)$.
(d) ${u} \cdot {v} =0$.
(e)${u} \cdot {u}=3$.
(f) ${v} \cdot {v}=4$.
Work Step by Step
(a) $\|u\|=\sqrt{3}, \quad \|v\|=2$.
(b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\left( -\frac{1}{2}, \frac{\sqrt{2}}{2},\frac{1}{2} \right)$.
(c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{\sqrt{3}}{3}\left(0, 1, \sqrt{2}\right)$.
(d) ${u} \cdot {v} =0$.
(e)${u} \cdot {u}=3$.
(f) ${v} \cdot {v}=4$.