Vectors Question (Orthogonality)

[haux]

Homework Statement

Let u1 = [2 -2 2 ] u2 = [-2 2 1], u3 = [0 1 2]Use the Gram-Schmidt process to u1, u2, u3, in this order. The resulting vectors are:

v1 = [___ ___ ___], v2 = [___ ___ ___], v3 = [___ ___ ___]

And ß = {v1, v2, v3} is an orthongal basis for R³.

Homework Equations

v1 = u1

v2 = u2 - u2 . v1 v1
------------ v1 . v1
v3= u3 - u3 . v1 v1 - u3 . v2 v2
---------------v1 . v1 v2 . v2

The Attempt at a Solution

Well, the first one is easy.

v1 = [2 -2 2]

I solved v2 and got [-1, 1, 2] which was also right.

I tried v3 based on the equation and got [1, 2/3, 4/3] which was apparently incorrect.

 

[HallsofIvy]

Why do you say "apparently" incorrect? You should have checked yourself that <2, -2, 2].[1, 2/3, 4/3]= 2- 4/3+ 8/3= 2+ 4/3 and [-1, 1, 2].[1, 2/3, 4/3]= -1+ 2/3+ 8/3= -1+ 10/3, neither of which is 0.

If you show what you did, that is show use exactly how you used that formula, we might be able to point out an error.

 

[Architeuthis Dux]

To find v3, we can use the same process as before, but this time using v2 instead of v1.

v3 = u3 - u3 . v2 v2
---------------v2 . v2

Substituting in the given values, we get:

v3 = [0 1 2] - [0 1 2] . [-1 1 2] [-1 1 2]
-------------------------- [-1 1 2] . [-1 1 2]

Simplifying, we get:

v3 = [0 1 2] - [-3] [-1 1 2]
-------------------------- [-6]

v3 = [0 1 2] + [3 3 -6]
------------------- [-6]

v3 = [3 4 -4]
--------- [-6]

v3 = [-1 -2/3 2/3]

Therefore, the resulting vectors are:

v1 = [2 -2 2]
v2 = [-1 1 2]
v3 = [-1 -2/3 2/3]

And ß = {v1, v2, v3} is an orthogonal basis for R3.