Dot products: Difference between revisions
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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'' is the inner product in geometric algebra. The dot product and its antiproduct are important for the calculation of angles and [[Geometric norm | norms]]. | ||
== Dot Product == | == Dot Product == | ||
The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{ | The dot product between two elements $$\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{ | :$$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ , | ||
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. | 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. | 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 | The antidot product can also be derived from the dot product using the De Morgan relationship | ||
:$$\mathbf a \mathbin{\unicode{ | :$$\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}$$ with itself. All other dot products and antidot products are zero. | |||
: | [[Image:Dots.svg|720px]] | ||
== In the Book == | |||
* The dot product and antidot product are introduced in Section 2.9. | |||
== See Also == | == See Also == | ||
Latest revision as of 01:27, 8 July 2024
The dot product is the inner product in geometric algebra. The dot product and its antiproduct are important for the calculation of angles and norms.
Dot Product
The dot product between two elements $$\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.