Show that $x,y \text{ and } z$ are linear independent iff $x \times y, x \times z \text{ and } y \times z$ are linear independent

Question

Let $ x,y,z \in{\mathbb R}^3$. Show that $x,y \text{ and } z$ are linear independent iff $x \times y, x \times z \text{ and } y \times z$ are linear independent. Where $\times$ denotes the cross product in ${\mathbb R}^3.$

So far I have only been able to show the forward implication: If $x,y,z$ are linear dependent, then $x \times y, x \times z \text{ and } y \times z$ are also linear dependent.

I need help with the backwards implication: If $x \times y, x \times z \text{ and } y \times z$ are linear independent, then $x,y,z$ are linear independent

I would be grateful for some help.

Answer 1

Assume the cross products are linearly independent. Suppose $ax+by+cz=0$ so $ax\times y + cz\times y =0$. Hence $a=c=0$. Similarly you can prove $b=0$. Since no non-zero choices of $a,\,b,\,c$ work, $x,\,y,\,z$ are linearly independent.

Answer 2

\begin{eqnarray*} det \left( \begin{array}{ccc} a_2 b_3-a_3 b_2 & a_3 b_1-a_1 b_3 & a_1 b_2-a_2 b_1 \\ b_2 c_3-b_3 c_2 & b_3 c_1-b_1 c_3 & b_1 c_2-b_2 c_1 \\ c_2 a_3-c_3 a_2 & c_3 a_1-c_1 a_3 & c_1 a_2-c_2 a_1 \\ \end{array} \right)=\left( det \left( \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{array} \right) \right)^2 \end{eqnarray*}