Answer
$\{(\frac{\sqrt 5}{5},\frac{2\sqrt 5}{5}),(\frac{-2\sqrt 5}{5},\frac{\sqrt 5}{5})\}$
Work Step by Step
Let $v_1=(1,2)\\
v_2=(-4,2)$
Check: $(v_1,v_2)=((1,2),(-4,2))=1.(-4)+2.2=0$
Hence, $v_1$ and $v_2$ are orthogonal vectors in $R^2$
To determine an orthogonal set, we obtain:
$$\frac{v_1}{||v_1||}=\frac{(1,2)}{||(1,2)||}=\frac{(1,2)}{\sqrt 1^2+2^2}=\frac{(1,2)}{\sqrt 5}=(\frac{\sqrt 5}{5},\frac{2\sqrt 5}{5})$$
$$\frac{v_2}{||v_2||}=\frac{(-4,2)}{||(-4,2)||}=\frac{(-4,2)}{\sqrt (-4)^2+2^2}=\frac{(-4,2)}{\sqrt 20}=(\frac{-2\sqrt 5}{5},\frac{\sqrt 5}{5})$$
Hence, a corresponding orthonormal set of vector is $\{(\frac{\sqrt 5}{5},\frac{2\sqrt 5}{5}),(\frac{-2\sqrt 5}{5},\frac{\sqrt 5}{5})\}$
