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


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


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


:$$\mathbf e_1 \mathbin{\unicode{x25CF}} \mathbf e_1 = 1$$
The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is nonzero only if they have the same grade.


:$$\mathbf e_2 \mathbin{\unicode{x25CF}} \mathbf e_2 = 1$$
== Antidot Product ==


:$$\mathbf e_3 \mathbin{\unicode{x25CF}} \mathbf e_3 = 1$$
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


:$$\mathbf e_4 \mathbin{\unicode{x25CF}} \mathbf e_4 = 0$$
:$$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,


:$$\mathbf e_i \mathbin{\unicode{x25CF}} \mathbf e_j = 0$$, for $$i \neq j$$.
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 dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is nonzero only if they have the same grade.
The antidot product can also be derived from the dot product using the De Morgan relationship


The following Cayley table shows the dot products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.
:$$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode["segoe ui symbol"]{x2022}} \underline{\mathbf b}}$$ .


== Table ==


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


== 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 $$\mathbf b$$".
[[Image:Dots.svg|720px]]


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


== In the Book ==


[[Image:AntidotProduct.svg|720px]]
* The dot product and antidot product are introduced in Section 2.9.


== See Also ==
== See Also ==

Latest revision as of 01:27, 8 July 2024

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.

See Also

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