need orthogonal basis for R3. i'm given one of them. how do i find the rest? I need orthogonal basis for R3. I am given v1 = (1,1,1), so I need so I need other two vectors in this basis but how do i find the other two?

at first i thought i would use gram schmidt but that doesn't seem plausible with just one vector.
Pick any vector $\mathbf u$ that is not a multiple of $\mathbf v_1$.
Set $\mathbf v_2 = \mathbf v_1 \times \mathbf u$.
Set $\mathbf v_3 = \mathbf v_1 \times \mathbf v_2$.
Just pick two more vectors arbitrarily and apply Gram-Schmidt. Gram-Schmidt preserves the direction of the first input vector, so you'll get a normalized copy of $(1, 1, 1)$.