Orthogonal basis

Could anyone tell me if any of the following sets of vectors is an orthogonal basis for R³

{(1,0,0), (1,1,0), (0,0,1)}
{(1,0,0), (1,-1,0), (0,0,1)}
{(1,0,1), (1,0,-1), (0,1,0)}
{(1,0,0), (1,1,0), (1,1,1)}

First of all the set of vectors have to be linearly independent, second the dot product of each pair of vectors must be equal to 0.

Can you first of all list all the set of vectors which are lin. independent ?

It's easy to eliminate three of the options by checking if they're orthogonal:

{(1,0,0), (1,1,0), (0,0,1)}:


{(1,0,0), (1,-1,0), (0,0,1)}:


{(1,0,1), (1,0,-1), (0,1,0)}:
This one is orthogonal:



To tell if the vectors are independent, set . Then:



and you can see the only solution is , which means the vectors are linearly independent.

{(1,0,0), (1,1,0), (1,1,1)}:


So the third set is an orthogonal basis, and the others aren't.

- Hollywood