Answer
(a) The set is orthogonal.
(b) $$u_1= \frac{1}{\sqrt{29}}(2,-5),$$ $$v_1=\frac{1}{\sqrt{116}}(10,4).$$ Now, the set $\{u_1,v_1\}$ is an orthonormal set.
Work Step by Step
Let $u=(2,-5)$ and $v=(10,4)$; then we have
(a) Since $$u\cdot v=20-20=0,$$ then the set is orthogonal.
(b) To normalize the set, we have $$u_1=\frac{1}{\|u\|}u=\frac{1}{\sqrt{29}}(2,-5),$$ $$v_1=\frac{1}{\|v\|}v=\frac{1}{\sqrt{116}}(10,4).$$ Now, the set $\{u_1,v_1\}$ is an orthonormal set.