Find an orthonormal basis of the plane x1 3x2 x3 0 Soluti

Find an orthonormal basis of the plane x_1 + 3x_2 - x_3 = 0.

Solution

x1 +3x2 -x3 =0

let the vectors which form the basis for this plane be u1 = (1 , 0, 1) and u2 = (0, 1 , 3)

Next we apply Gram-Schmidt to this basis to make it orthonormal.

v1 = u1/||u1|| = (1, 0,1)/sqrt2

v2 = u1 - (v1.u2)v1 = (1 , 0 , 1) - (3/sqrt2)(1, 0,1)/sqrt2

= (1, 0 , 1) - (3/2, 0 , 3/2)

= ( -1/2 , 0 , -1/2)

Orthonormal basis = ( -1/sqrt2 , 0, -1/sqrt2) ,   (1, 0,1)/sqrt2