Orthonormal Basis of a Plane

1. May 24, 2006

gbacsf

I need to find the Orthonormal Basis of this plane:

x - 4y -z = 0

I know the result will be the span of two vectors but I'm not sure where to start. Any hints?

Thanks,

Gab

2. May 24, 2006

AKG

First find a basis by finding two independent vectors that satisfy that equation. This is easy: find one non-zero vector satisfying that equation with z-component 0, and find another satisfying that equaiton with y-componenet 0. Next, orthogonalize this basis using Gramm-Schmidt. Finally, normalize it by dividing the two orthogonal vectors you have by their own norms.

3. May 24, 2006

gbacsf

So set (y=1, z=0) and (y=0, z=1)

Get two vectors:

(4,1,0) and (1,0,1)

Normalize:

(4/sqrt(17), 1/sqrt(17), 0) and (1/sqrt(2), 0, 1/sqrt(2))

4. May 24, 2006

Integral

Staff Emeritus
To satisfy the "ortho" part of Orthonormal you need to verify that the dot product of your 2 vectors is 0.

5. May 24, 2006

gbacsf

Ah thanks,

so e1= (1/sqrt(2), 0, 1/sqrt(2))

e2 = (2/3, 1/3, -2/3)