Dot products - Revision history

Eric Lengyel at 01:27, 8 July 2024

2024-07-08T01:27:26Z

← Older revision		Revision as of 01:27, 8 July 2024
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	== 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 is defined as		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

	:$$\mathbf a \mathbin{\unicode{~~x25CF~~}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,

	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.
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	== Antidot Product ==		== 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		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

	:$$\mathbf a \mathbin{\unicode{~~x25CB~~}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,

	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]].		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		The antidot product can also be derived from the dot product using the De Morgan relationship

	:$$\mathbf a \mathbin{\unicode{~~x25CB~~}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{~~x25CF~~}} \underline{\mathbf b}}$$ .		:$$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode["segoe ui symbol"]{x2022}} \underline{\mathbf b}}$$ .

	== Table ==		== Table ==

Eric Lengyel at 23:26, 13 April 2024

2024-04-13T23:26:29Z

← Older revision		Revision as of 23:26, 13 April 2024
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	[[Image:Dots.svg\|720px]]		[[Image:Dots.svg\|720px]]


			== In the Book ==

			* The dot product and antidot product are introduced in Section 2.9.

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

Eric Lengyel at 01:53, 13 April 2024

2024-04-13T01:53:48Z

← Older revision		Revision as of 01:53, 13 April 2024
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	:$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,

	where $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, $$\mathbf G$$ 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.
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	:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,

	where, again, $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, but now $$\mathbb G$$ is the $$16 \times 16$$ metric antiexomorphism matrix.		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		The antidot product can also be derived from the dot product using the De Morgan relationship

Eric Lengyel at 22:02, 21 January 2024

2024-01-21T22:02:14Z

← Older revision		Revision as of 22:02, 21 January 2024
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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 ==
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	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.

	~~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 ==


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

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

	where, again, $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, but now $$\mathbb G$$ is the $$16 \times 16$$ metric ~~anti-exomorphism~~ matrix.		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		The antidot product can also be derived from the dot product using the De Morgan relationship
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	:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x25CF}} \underline{\mathbf b}}$$ .		:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \overline{\underline{\mathbf a} \mathbin{\unicode{x25CF}} \underline{\mathbf b}}$$ .

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

	~~[[Image:AntidotProduct.svg\|720px]]~~

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

Eric Lengyel at 07:35, 23 October 2023

2023-10-23T07:35:22Z

← Older revision		Revision as of 07:35, 23 October 2023
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	:$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,

	where $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, $$\mathbf G$$ is the $$16 \times 16$$ ~~extended~~ metric ~~tensor~~, 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.
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	:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,

	where, again, $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, but now $$\mathbb G$$ is the $$16 \times 16$$ ~~extended antimetric tensor~~.		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 anti-exomorphism matrix.

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

Eric Lengyel at 18:29, 25 August 2023

2023-08-25T18:29:13Z

← Older revision		Revision as of 18:29, 25 August 2023
Line 7:		Line 7:
	:$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b = \mathbf a^{\mathrm T}\mathbf G \mathbf b$$ ,

	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.		where $$\mathbf a$$ and $$\mathbf b$$ are treated as generic vectors over the 16 basis elements of the algebra, $$\mathbf G$$ is the $$16 \times 16$$ extended metric tensor, 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.
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	:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,		:$$\mathbf a \mathbin{\unicode{x25CB}} \mathbf b = \mathbf a^{\mathrm T}\mathbb G \mathbf b$$ ,

	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.		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 $$16 \times 16$$ extended antimetric tensor.

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

Eric Lengyel at 18:28, 25 August 2023

2023-08-25T18:28:10Z

← Older revision		Revision as of 18:28, 25 August 2023
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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 $$\mathcal G_{3,0,1}$$.

Eric Lengyel: 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..."

2023-07-15T06:26:19Z

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 and antidot product are important for the calculation of [[Geometric norm | 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 includes the metric properties of the [[geometric product]], which means we have the following rules for the basis vectors.

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

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

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

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

:$$\mathbf e_i \mathbin{\unicode{x25CF}} \mathbf e_j = 0$$, for $$i \neq j$$.

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 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 $$\mathbf b$$".

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

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

== See Also ==

* [[Geometric products]]
* [[Wedge products]]