Cross product and dot product

Question

What's the easiest way to understand and prove that $A \cdot B \times C = C \cdot A \times B $ ?

Answer 1

You might make use of the fact that for $A,B,C \in \mathbb R^3$, $A \cdot (B \times C) = (A \times B) \cdot C = \det M$, where $M$ is the matrix made from the column vectors $A, B, C$. Both identities follows from the Sarrus formula for determinants of $3\times 3$ matrices.

Answer 2

Show that the first is the determinant of the matrix whose rows are $A,B,C$.

Answer 3

For $A = (a_1, a_2, a_3)$, $B = (b_1, b_2, b_3$, an $C= (c_1, c_2, c_3)$ you could simply just compute each side manually, so you would for example get for the left hand side:

$$\begin{align} A\cdot B\times C &= (a_1, a_2, a_3)\cdot (b_2c_3 - b_3c_2, b_3c_1 - b_1c_3, b_1c_2 - b_2c_1) \\ &= ... \end{align} $$

And then you compute the right hand side and check that you got the same thing.

Q: Is this the easiest way? A: Probably not, but it might be a good exercise in keeping track of terms.

Added: As for the understanding. The quantity $A\cdot B\times C$ is called a triple product. If the three vectors are not in the same plane, then they span a parallelepiped, and the absolute value of the triple product gives the volume of the the parallelepiped.