Answer
(a) $\|u\|=\frac{3\sqrt{209}}{40}, \quad \|v\|=\frac{\sqrt{41}}{20}$.
(b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\frac{20\sqrt{41}}{41}\left(0, \frac{1}{4}, \frac{1}{5}\right)$.
(c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{40\sqrt{209}}{627}\left(1, \frac{1}{8}, \frac{2}{5}\right)$.
(d) ${u} \cdot {v} =\frac{89}{900}$.
(e)${u} \cdot {u}=\frac{1881}{1600}$.
(f) ${v} \cdot {v}=\frac{41}{400}$.
Work Step by Step
(a) $\|u\|=\frac{3\sqrt{209}}{40}, \quad \|v\|=\frac{\sqrt{41}}{20}$.
(b) Unit vector in direction of $v$ is $\frac{v}{\|v\|}=\frac{20\sqrt{41}}{41}\left(0, \frac{1}{4}, \frac{1}{5}\right)$.
(c) Unit vector in direction opposite to $u$ is $-\frac{u}{\|u\|}=-\frac{40\sqrt{209}}{627}\left(1, \frac{1}{8}, \frac{2}{5}\right)$.
(d) ${u} \cdot {v} =\frac{89}{900}$.
(e)${u} \cdot {u}=\frac{1881}{1600}$.
(f) ${v} \cdot {v}=\frac{41}{400}$.
