# What is the geometric interpretation of the vector triple product?

The interpretation of the vector product is the area of the parallelogram with sides made up of a and b and the scalar triple product is the volume of the parallelpiped with sides a, b, and c, but what is the interpretation of the vector triple product. Is it just simply the area of the parallelogram with sides p and c, where p = a x b, or is it something else that can't really be visualized?

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SteamKing
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The interpretation of the vector product is the area of the parallelogram with sides made up of a and b and the scalar triple product is the volume of the parallelpiped with sides a, b, and c, but what is the interpretation of the vector triple product. Is it just simply the area of the parallelogram with sides p and c, where p = a x b, or is it something else that can't really be visualized?

The triple product Ax(BxC) is another vector:

The vector triple product, Ax(BxC) is a vector, is normal to A and normal to BxC which means it is in the plane of B and C. And it is linear in all three vectors.
http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter05/section06.html

A few observations I find interesting:
• Note that the answer is completely independent of the right hand rule: the left hand rule would give just the same answer. This is a good clue that we're dealing with a real vector, and not a pseudovector (which is really a disguised bivector).
• The cross product only makes sense in 3 dimensions, but the "vector triple product" makes perfect sense in arbitrary numbers of dimensions. If we're in more than 3 dimensions, the 3 vectors involved actually form a basis for their own 3-dimensional subspace. The cross product you'd use is the one defined in that subspace. Neat!

To get a geometric interpretation, we can rewrite this in geometric algebra. We'll end up with a more direct formula, which incidentally uses no cross products at all.

Here is the double cross product rewritten in geometric algebra (derivation omitted):
$$a \times (b \times c) = -a \rfloor (b \wedge c)$$
Let's break this down.

$(b \wedge c)$ is a "bivector"; call it $B$. It's an area element in the $bc$ plane, oriented from $b$ to $c$.

$a \rfloor B$ is the "left contraction" of $a$ onto $B$ -- kind of like a dot product. Basically, it does the following:
1. Projects $a$ onto the plane of $B$
2. Rotates it 90 degrees in the direction of $B$, i.e., from $b$ to $c$
And of course, there's a minus sign, so the net effect is like a 90 degree rotation in the opposite sense.

So that's what $a \times (b \times c)$ means: project $a$ onto the $bc$ plane, then rotate it 90 degrees in the direction from $c$ to $b$. Nice and direct -- no fumbling with awkward right hand (or left hand!) rules.