Understanding the Meaning of (e1^e2)\cdote3 in Geometric Algebra

[JakeD]

What is the meaning of (e₁^e₂)$$\cdot$$e₃?

(outer product multiplied by inner product)

 

[Some Pig]

The volume of the parallelepiped formed by those vectors.

 

[JakeD]

Thank you.

Jake

 

[chogg]

No, this is just wrong! The volume of the parallelepiped would be all outer products, $e_1 \wedge e_2 \wedge e_3$.

$(e_1 \wedge e_2)$ is a bivector. So you're asking: what is the inner product of a vector with a bivector? That has a clear geometrical interpretation. For a vector $a$ and bivector $B$, $a \cdot B$ does the following:

1. Project $a$ onto the plane defined by $B$
2. Rotate 90 degrees in the "sense" of $B$
3. Dilate by the magnitude of $B$

Note that this uses all three defining characteristics of the bivector $B$:

• Attitude (basically the angle the plane makes in space)
• Orientation (clockwise vs. counterclockwise)
• Magnitude (i.e. area)

With the inner product used by Hestenes et al, you also have
$$a \cdot B = - B \cdot a$$
which let's you answer your question.

By the way, in 3D, your construction is equivalent to the "double cross product" (not the "vector triple product"):
$$(e_1 \wedge e_2) \cdot e_3 = - (e_1 \times e_2) \times e_3$$
Note how the GA version (described above) is much more intuitive and easy to visualize -- the VA version (double cross product) will give you carpal tunnel from all those applications of the right-hand rule!

 

[JakeD]

Thanks chogg for your details answer; I later noticed indeed that his answer is wrong.
GAViewer also demonstrates this very nicely.