Dot products: Difference between revisions

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(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...")
 
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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.
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 and antidot product are important for the calculation of [[Geometric norm | norms]].


== 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


The dot product includes the metric properties of the [[geometric product]], which means we have the following rules for the basis vectors.
:$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,


:$$\mathbf e_1 \mathbin{\unicode{x25CF}} \mathbf e_1 = 1$$
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.


:$$\mathbf e_2 \mathbin{\unicode{x25CF}} \mathbf e_2 = 1$$
The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is nonzero only if they have the same grade.


:$$\mathbf e_3 \mathbin{\unicode{x25CF}} \mathbf e_3 = 1$$
The following Cayley table shows the dot products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.


:$$\mathbf e_4 \mathbin{\unicode{x25CF}} \mathbf e_4 = 0$$


:$$\mathbf e_i \mathbin{\unicode{x25CF}} \mathbf e_j = 0$$, for $$i \neq j$$.
[[Image:DotProduct.svg|720px]]


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


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


:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,


[[Image:DotProduct.svg|720px]]
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.


== Antidot product ==
The antidot product can also be derived from the dot product using the De Morgan relationship


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 $$\mathbf b$$".
:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x25CF}} \underline{\mathbf b}}$$ .


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 $$\mathcal G_{3,0,1}$$.


File:DotProduct.svg

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 $$\mathcal G_{3,0,1}$$.


File:AntidotProduct.svg

See Also