Dot products

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The dot product is the inner product in geometric algebra, and it makes up the scalar part of the geometric product. There are two products with symmetric properties called the dot product and antidot product.

Dot Product

The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b$$ and read "$$\mathbf a$$ dot $$\mathbf b$$".

The dot product includes the metric properties of the geometric product, which means we have the following rules for the basis vectors.

$$\mathbf e_1 \mathbin{\unicode{x25CF}} \mathbf e_1 = 1$$
$$\mathbf e_2 \mathbin{\unicode{x25CF}} \mathbf e_2 = 1$$
$$\mathbf e_3 \mathbin{\unicode{x25CF}} \mathbf e_3 = 1$$
$$\mathbf e_4 \mathbin{\unicode{x25CF}} \mathbf e_4 = 0$$
$$\mathbf e_i \mathbin{\unicode{x25CF}} \mathbf e_j = 0$$, for $$i \neq j$$.

The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is nonzero only if they have the same grade.

The following Cayley table shows the dot products between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.


DotProduct.svg

Antidot product

The antidot product is a dual to the dot product. The antidot product between elements $$\mathbf a$$ and $$\mathbf b$$ is often written $$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b$$ and is read as "$$\mathbf a$$ antidot $$\mathbf b$$".

The following Cayley table shows the antidot products between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.


AntidotProduct.svg

See Also