Prove that $(a ∧ b)\cdot c\ne0\iff a, b$ and $c$ are linearly independent, where $a,b,c\in\Bbb R^3$.
In the first part of the question, I proved that $(a ∧ b)$ is orthogonal to $a$ and to $b$, so I'm thinking this may come in useful?
I have worked out the following: $$(a ∧ b)\cdot c = a_1(a_2b_3-b_2a_3) + a_2(a_3b_1-b_3a_1) + a_3(a_1b_2-b_1a_2)$$ where $a = (a_1,a_2,a_3)$ etc.
I've also been able to prove that if they are not linearly independent (i.e. they are scalar multiples of one another), then $(a ∧ b)\cdot c = 0$, but I'm really stuck on what to do now?
Thanks in advance!! :)
$a,b,c$ can be written in terms of the unit vectors $e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$. They span exactly the same space, and therefore $(a\wedge b)\cdot c$ is the exactly the same as for the $e_1,e_2,e_3$ representation, which is never zero.