Question

find an orthonormal basis of the plane x1-8x2-x3=0

Answer 1

You just pick one vector which is in the plane, for example (1,0,1), then scale it to unit length, so you get \[e_1(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}).\] To find e_2 use Gram-Schmidt method to another vector, for example v=(0,1/8,-1).
\[e_2=v-\frac{e_1\cdot v}{e_1\cdot e_1} e_1=(0,1/8,-1)-(-1/\sqrt{2})(1/\sqrt{2},0,1/\sqrt{2})\] which is equal to \[e_2=(1/2,1/8,-1/2)\] after scaling to unit length you'll get \[\hat{e}_2=\frac{\sqrt{64}}{\sqrt{33}}(1/2,1/8,-1/2)\] e_1 and \hat{e}_2 form an orthonormal basis to the plane.