Answer
(a) the set is orthogonal.
(b) The set $\{u_1,v_1\}$
$$u_1= (-\frac{2}{3}, \frac{1}{3},\frac{2}{3})$$
$$v_1= =(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}},0).$$
is orthonormal set.
Work Step by Step
Let $u=(-\frac{2}{15}, \frac{1}{15},\frac{2}{15})$ and $v=(\frac{1}{15}, \frac{2}{15},0)$, then we have
(a) since $$u\cdot v=- \frac{2}{225}+ \frac{2}{225}=0,$$
then the set is orthogonal.
(b) to normalize the set, we have
$$u_1=\frac{1}{\|u\|}u=\frac{15}{3}(-\frac{2}{15}, \frac{1}{15},\frac{2}{15})=(-\frac{2}{3}, \frac{1}{3},\frac{2}{3})$$
$$v_1=\frac{1}{\|v\|}v=\frac{15}{\sqrt{5}}(\frac{1}{15}, \frac{2}{15},0)=(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}},0)$$
Now, the set $\{u_1,v_1\}$ is orthonormal set.
