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| The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct.
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| == Geometric Product ==
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| The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct are present in the same context, which is now known to be necessary for a proper understanding of the algebra. To remedy the situation, we write the geometric product between elements $$\mathbf a$$ and $$\mathbf b$$ as $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ and read it as "$$\mathbf a$$ wedge-dot $$\mathbf b$$".
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| The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{3,0,1}$$ means that three basis vectors square to +'''1''', zero basis vectors square to −'''1''', and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.
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| :$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = \mathbf 1$$
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| :$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = \mathbf 1$$
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| :$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = \mathbf 1$$
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| :$$\mathbf e_4 \mathbin{\unicode{x27D1}} \mathbf e_4 = 0$$
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| :$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.
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| The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.
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| Cells colored yellow correspond to the contribution from the [[wedge product]].
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| [[Image:GeometricProduct.svg|720px]]
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| == Geometric Antiproduct ==
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| The geometric antiproduct is a dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as "$$\mathbf a$$ antiwedge-dot $$\mathbf b$$".
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| The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.
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| :$$\overline{\mathbf e_1} \mathbin{\unicode{x27C7}} \overline{\mathbf e_1} = {\large\unicode{x1D7D9}}$$
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| :$$\overline{\mathbf e_2} \mathbin{\unicode{x27C7}} \overline{\mathbf e_2} = {\large\unicode{x1D7D9}}$$
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| :$$\overline{\mathbf e_3} \mathbin{\unicode{x27C7}} \overline{\mathbf e_3} = {\large\unicode{x1D7D9}}$$
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| :$$\overline{\mathbf e_4} \mathbin{\unicode{x27C7}} \overline{\mathbf e_4} = 0$$
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| :$$\overline{\mathbf e_i} \mathbin{\unicode{x27C7}} \overline{\mathbf e_j} = \overline{\mathbf e_i} \vee \overline{\mathbf e_j}$$, for $$i \neq j$$.
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| The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.
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| Cells colored yellow correspond to the contribution from the [[antiwedge product]].
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| [[Image:GeometricAntiproduct.svg|720px]]
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| == De Morgan Laws ==
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| There are many possible geometric products and antiproducts. The signs of the results they produce differ in grade-dependent ways, but are otherwise equivalent. The relationship between the product and antiproduct is fixed by a specific choice of dualization function that exchanges full and empty dimensions. We choose the left and right [[complements]] as the dualization function and its inverse. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.
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| :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$
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| :$$\overline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \overline{\mathbf b}$$
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| :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \underline{\mathbf b}$$
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| :$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$
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| == See Also ==
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| * [[Wedge products]]
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| * [[Dot products]]
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| * [[Complements]]
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