## anonymous 5 years ago find an orthonormal basis of the plane x1-8x2-x3=0

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.