I'm having trouble understanding how the volume of a parallelopipede is calculated to consistent values using vectors; the Wikipedia page for the triple product says that the operation is basically invariant but makes a point of saying that you can't just switch around vectors without changing the value, (specifics below)

Question

mendicant_bias · Answer

Let's say we have vectors A, B, and C; none are parallel. If we calculate the triple scalar product,
$$a  * (b \times c)$$

anonymous · Answer

The triple scalar product, $\vec a \cdot (\vec b \times \vec c)$, is a practice that can be used to take the volume of any parallelepiped, including prisms, cubes, whatever. Wikipedia has a pretty clear proof on it.

mendicant_bias · Answer

We can freely rotate the terms as such:
$$a * (b \times c) = b * (c \times a) = c * (a \times b)$$
However, we can't switch the values in the cross product without changing the final value. How, if these vectors are seemingly arbitrarily selected, do we get the same outcome? 

I might just be misinterpretaing the WIki page, but to my knowledge, 
$$a * (b \times c) \neq c * (b \times a)$$

mendicant_bias · Answer

Yeah, I'm not doubting that it can be used in this manner in the slightest. I'm confused as to whether it's completely algebraically invariant, because if it isn't, and I don't think it is from what I've read, it would seem like arbitrarily shifting around any of the vectors in the scalar triple product equation would give different results.

mendicant_bias · Answer

(And I get that the RHS of the last eqn could be considered the negative of the third scalar triple product, but then I'm wondering "which one is correct?")

ganeshie8 · Answer

$V = a * (b \times c) = b * (c \times a) = c * (a \times b)$

$-V =  a * (c \times b) = b * (a \times c) = c * (b \times a)$

ganeshie8 · Answer

both are right, just to do wid right handed / left handed coordinate system

ganeshie8 · Answer

above both are valid for Right handed system

mendicant_bias · Answer

IDK what right/left handed coordinate systems are, I've never heard of the term; I'll go read some on this and then come back. Thanks for the clarification, I've gotta go.

anonymous · Answer

Right-handed system is where $\hat i \times \hat j = \hat k$.

anonymous · Answer

Try messing around with the triple product identities, like this: http://i.imgur.com/S5w2fZK.jpg

anonymous · Answer

Some things are more fun to discover.