Find an orthonormal basis of the plane x14x2x30Solutionx1 4x

Find an orthonormal basis of the plane x14x2x3=0

x1 -4x2 -x3 =0

one possible basis of the plane consisiting of v1= ( 0,-1, 4) and v2 = ( 2, 0,2)

u1 = v1/||v1|| = (0,-1,4)/sqrt17

u2 = v2 - (u1.v2)u1 = (2,0,2) - (8/sqrt17)(0,-1,4)/sqrt17

=( 2,8 , 30)/sqrt968

So, u1 and u2 form the orthonormal basis