Find all vectors $v = (x,y,z)$ orthogonal to both $u_1$ and $u_2$.

Question

$u_1 = (2, -1, 3)$ and $u_2 = (0, 0, 0)$

I tried using the cross product of the two but that just gave me the zero vector. I don't know any other methods to get a vector that is orthogonal to two vectors.

The answer is $v = s(1, 2, 0) + t(0, 3, 1)$ , where $s$ and $t$ are scalar values.

Answer 1

Remember that every vector is orthogonal to the 0 vector. So, we are really only looking for all vectors orthogonal $\langle 2,-1,3\rangle$. This is a plane, through the origin with normal vector $\langle 2,-1,3\rangle$.

Answer 2

Every vector is orthogonal to $(0,0,0)$ as the dot product with it is zero. You just need the dot product with $(2,-1,3)$ to be zero as well. If the vector is $(x,y,z)$ you need $2x-y+3z=0$, which is a plane. You need to find two vectors in that plane and they will span the space of interest. Any two that are not colinear will do.