# Homework Help: Orthogonal complement of the intersection of 2 planes

1. May 2, 2015

### fattycakez

1. The problem statement, all variables and given/known data
Let W be the intersection of the two planes: x-y+z=0 and x+y+z=0
Find a basis for and the dimension of the orthogonal complement, W

2. Relevant equations

3. The attempt at a solution
The line x+z=0 intersects the plane, which is parameterized as t(1, 0, -1)
Then W is the plane x-z=0
Then the nullspace of this plane is (1, 0, 1) which is the basis for W
And the dimension is 1?
Am I even in the right ballpark here? :D

2. May 2, 2015

### haruspex

You need a constraint on y as well to make it a line (but you got that right in the parametric form).
Correct.
The only context I know for the term null space is in connection with transformations, and there is no transformation being discussed here.
The vector you state is a basis for W.

3. May 2, 2015

### fattycakez

Okay sweet!
So if (1,0,1) is the basis for W, shouldn't there be one more basis vector since W is a plane and a plane is 2 dimensional?

4. May 2, 2015

### haruspex

No, I wrote that it is a basis for W, not W.

5. May 2, 2015

### fattycakez

Okay so how do you find the basis for W then?

6. May 2, 2015

### haruspex

It's not 'the' basis, it's 'a' basis.
You correctly stated the constraint for it, x=z. You just need two independent vectors in it.