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.
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 $\mathbf v_3 = \mathbf v_1 \times \mathbf v_2$.
Take for example:
$(1,-1,0)$
and
$(1,1,-2)$
The "standard" way is to find two more vectors then use Gram-Schmidt, as suggested by others. For a short cut using trial and (not much) error, take $(1,-1,0)$ which is orthogonal to your first vector. Then $(1,1,a)$. This is orthogonal to the second vector, and also to the first if you choose $a$ suitably. I expect this is how Yiorgos obtained his answer.
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 $(1, 1, 1)$.