Orthogonal complement of the intersection of 2 planes

[fattycakez]

Homework Statement

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^⊥

Homework Equations

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

 

[haruspex]

The line x+z=0

You need a constraint on y as well to make it a line (but you got that right in the parametric form).

Then W^⊥ is the plane x-z=0

Correct.

Then the nullspace of this plane is (1, 0, 1) which is the basis for W^⊥

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.

 

[fattycakez]

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.

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?

 

[haruspex]

So if (1,0,1) is the basis for W^⊥

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

 

[fattycakez]

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

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

 

[haruspex]

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

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.