Dot products: Difference between revisions
Eric Lengyel (talk | contribs) (Created page with "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. The dot product and antidot product are important for the calculation of norms. == Dot Product == The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b$$ and r...") |
Eric Lengyel (talk | contribs) No edit summary |
||
| Line 1: | Line 1: | ||
The ''dot product'' is the inner product in geometric algebra | The ''dot product'' is the inner product in geometric algebra. The dot product its antiproduct are important for the calculation of angles and [[Geometric norm | norms]]. | ||
The dot product | |||
== Dot 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 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, and $$\mathbf G$$ is the $$16 \times 16$$ extended metric tensor. | |||
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 rigid geometric algebra $$\mathcal G_{3,0,1}$$. | |||
: | [[Image:DotProduct.svg|720px]] | ||
== Antidot product == | |||
The | 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, and $$\mathbb G$$ is the $$16 \times 16$$ extended antimetric tensor. | |||
The antidot product can also be derived from the dot product using the De Morgan relationship | |||
:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x25CF}} \underline{\mathbf b}}$$ . | |||
The following Cayley table shows the antidot products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. | The following Cayley table shows the antidot products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. | ||
Revision as of 18:28, 25 August 2023
The dot product is the inner product in geometric algebra. The dot product 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, and $$\mathbf G$$ is the $$16 \times 16$$ extended metric tensor.
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 rigid geometric algebra $$\mathcal G_{3,0,1}$$.
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, and $$\mathbb G$$ is the $$16 \times 16$$ extended antimetric tensor.
The antidot product can also be derived from the dot product using the De Morgan relationship
- $$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x25CF}} \underline{\mathbf b}}$$ .
The following Cayley table shows the antidot products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.
