Vector Product of complex vectors

Question

I need to get a vector orthogonal to $\mathbf{A}=(1,i,1,-i)$ and $\mathbf{B}=(i,1,i,-1)$ where $i^2=\sqrt{-1}$. Note also that $\mathbf{A}.\mathbf{B}=0$.

I was thinking to take the cross product, but not sure if this is possible/appropriate for vectors of length 4. Would appreciate any advice/comments.

Answer 1

You can find the set of all vectors orthogonal to $A$ and $B$ by setting up the linear system of two equation in four unknowns $X = [x_1, x_2, x_3, x_4] $ as follows

$ \begin{bmatrix} A^T \\ B^T \end{bmatrix} {X^*} = 0 $

Solving, you get

$ X = (-t, s,t, s) $ , $t,s \in \mathbb{R} $

Verify

$A \cdot X = (-t) + i s + t - i s = 0 $

$B \cdot X = -i t + s + i t - s = 0 $