Dot products
The dot product is the inner product in geometric algebra. The dot product and its antiproduct are important for the calculation of angles and norms.
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 is defined as
- $$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,
where $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, $$\mathbf G$$ is the $$16 \times 16$$ metric exomorphism matrix, and we are using ordinary matrix multiplication.
The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is nonzero only if they have the same grade.
Antidot Product
The antidot product between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b$$ and is read as "$$\mathbf a$$ antidot $$\mathbf b$$". The antidot product is defined as
- $$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,
where, again, $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, but now $$\mathbb G$$ is the $$16 \times 16$$ metric antiexomorphism matrix.
The antidot product can also be derived from the dot product using the De Morgan relationship
- $$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x25CF}} \underline{\mathbf b}}$$ .
Table
The following table shows the dot product and antidot product of each basis element $$\mathbf u$$ in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ with itself. All other dot products and antidot products are zero.
In the Book
- The dot product and antidot product are introduced in Section 2.9.