If $v$ is orthogonal to vectors $x$ and $y$, then $v$ is a scalar multiple of $x \times y$.
So far I have that:
$v\cdot x=v_1x_1+v_2x_2+v_3x_3=0$ and
$v\cdot y=v_1y_1+v_2y_2+v_3y_3=0$
$x \times y = (x_2y_3-x_3y_2,x_3y_1-x_1y_3,x_1y_2-x_2y_1)$
But I'm not sure how to show that $v$ is a scalar multiple of $x\times y$
Any tips on how I can proceed? Thanks!
The statement is trivial if $x$ and $y$ are linearly dependent, because then $x\times y=0$ and so $v=0.(v\times y)$.
Otherwise, $\dim\operatorname{span}(\{x,y\})=2$, and therefore $\dim\operatorname{span}(\{x,y\})^\perp=1$. So, since $v,x\times y\in\operatorname{span}(\{x,y\})^\perp$, and since $x\times y\ne0$, $v$ is a scalar multiple of $x\times y$.