# 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 $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$ and read "$$\mathbf a$$ dot $$\mathbf b$$". The dot product is defined as

- $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,

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 $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b$$ and is read as "$$\mathbf a$$ antidot $$\mathbf b$$". The antidot product is defined as

- $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,

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

- $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode["segoe ui symbol"]{x2022}} \underline{\mathbf b}}$$ .

## 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.