Here's my attempt:
Let $𝒙 = (x_1, x_2, x_3)$ and $𝒚 = (y_1, y_2, y_3)$
The cross product of $𝒙, 𝒚$ is $𝒙⨯𝒚=(x_2y_3-x_3y_2, x_3y_1 - x_1y_3, x_1y_2 - x_2y_1)$
And linear independence of $𝒙, 𝒚$ means that if $a𝒙 + b𝒚 = 0$, then $a = b = 0$, i.e. $a(x_1, x_2, x_3) + b(x_1, x_2, x_3) = 0 ⇒ a = b = 0$
So if cross product is nonzero, then $ x_2y_3-x_3y_2 + x_3y_1 - x_1y_3 + x_1y_2 - x_2y_1 ≠ 0, i.e. x_1(y_2-y_3)+x_2(y_3-y_1) + x_3(y_1 - y_2) ≠ 0$
And then I'm just getting confused.
I can't seem to connect the two in a formal proof.