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## Main Question or Discussion Point

Geometric Algebra for Physicists, in equation (4.56) introduces the following notation

[tex]

A * B = \langle AB \rangle

[/tex]

as well as (4.57) the commutator product:

[tex]

A \times B = \frac{1}{2}\left(AB - BA\right)

[/tex]

I can see the value defining the commutator product since this selects all the odd grade terms of a product (this can be used to express the othogonal component of a vector with respect to another (rejection), or the othogonal component of a plane with respect to an intersecting plane (or any plane in [tex]\mathbb{R}^3[/tex])).

But what's the point of introducing a second notation for grade zero selection? If one is going to introduce an operator to complement the commutator product, then something like the following would make more sense:

[tex]

A * B = \frac{1}{2}\left(AB + BA\right)

[/tex]

ie: select all the even grade components of the product. This is only equivalent to [tex]\langle AB \rangle[/tex] for specific cases like the symmetric vector product (dot product), for intersecting bivectors, for trivectors with bivector intersection, ...

Does anybody else think that this is probably a typo in the text? I haven't been reading this book linearly (skipping back and forth between it and Hestenes NFCM) so I haven't seen where or if either this commutator product or this * product are employed (seeing the usage would probably confirm if this is a typo).

[tex]

A * B = \langle AB \rangle

[/tex]

as well as (4.57) the commutator product:

[tex]

A \times B = \frac{1}{2}\left(AB - BA\right)

[/tex]

I can see the value defining the commutator product since this selects all the odd grade terms of a product (this can be used to express the othogonal component of a vector with respect to another (rejection), or the othogonal component of a plane with respect to an intersecting plane (or any plane in [tex]\mathbb{R}^3[/tex])).

But what's the point of introducing a second notation for grade zero selection? If one is going to introduce an operator to complement the commutator product, then something like the following would make more sense:

[tex]

A * B = \frac{1}{2}\left(AB + BA\right)

[/tex]

ie: select all the even grade components of the product. This is only equivalent to [tex]\langle AB \rangle[/tex] for specific cases like the symmetric vector product (dot product), for intersecting bivectors, for trivectors with bivector intersection, ...

Does anybody else think that this is probably a typo in the text? I haven't been reading this book linearly (skipping back and forth between it and Hestenes NFCM) so I haven't seen where or if either this commutator product or this * product are employed (seeing the usage would probably confirm if this is a typo).