From Rigid Geometric Algebra
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| 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]].
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| == Dot Product ==
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| 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
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| :$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,
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| 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.
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| The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is nonzero only if they have the same grade.
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| == Antidot Product ==
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| 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
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| :$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,
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| 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]].
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| The antidot product can also be derived from the dot product using the De Morgan relationship
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| :$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x25CF}} \underline{\mathbf b}}$$ .
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| == Table ==
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| 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.
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| [[Image:Dots.svg|720px]]
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| == See Also ==
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| * [[Geometric products]]
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| * [[Wedge products]]
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Latest revision as of 01:53, 13 April 2024