http://projectivegeometricalgebra.org/wiki/index.php?title=Special:NewPages&feed=atom&hideredirs=1&limit=50&offset=&namespace=0&username=&tagfilter=&size-mode=max&size=0 Projective Geometric Algebra - New pages [en] 2022-01-26T09:10:48Z From Projective Geometric Algebra MediaWiki 1.35.2 http://projectivegeometricalgebra.org/wiki/index.php?title=Dot_products Dot products 2021-10-16T21:01:01Z <p>Eric Lengyel: </p> <hr /> <div>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.<br /> <br /> == Dot Product ==<br /> <br /> The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b$$ and read &quot;$$\mathbf a$$ dot $$\mathbf b$$&quot;.<br /> <br /> The dot product includes the metric properties of the [[geometric product]], which means we have the following rules for the basis vectors.<br /> <br /> :$$\mathbf e_1 \mathbin{\unicode{x25CF}} \mathbf e_1 = 1$$<br /> <br /> :$$\mathbf e_2 \mathbin{\unicode{x25CF}} \mathbf e_2 = 1$$<br /> <br /> :$$\mathbf e_3 \mathbin{\unicode{x25CF}} \mathbf e_3 = 1$$<br /> <br /> :$$\mathbf e_4 \mathbin{\unicode{x25CF}} \mathbf e_4 = 0$$<br /> <br /> :$$\mathbf e_i \mathbin{\unicode{x25CF}} \mathbf e_j = 0$$, for $$i \neq j$$.<br /> <br /> The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is nonzero only if they have the same grade.<br /> <br /> The following Cayley table shows the dot products between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.<br /> <br /> <br /> [[Image:DotProduct.svg|720px]]<br /> <br /> == Antidot product ==<br /> <br /> 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 &quot;$$\mathbf a$$ antidot $$\mathbf b$$&quot;.<br /> <br /> The following Cayley table shows the antidot products between all pairs of basis elements in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$.<br /> <br /> <br /> [[Image:AntidotProduct.svg|720px]]<br /> <br /> == See Also ==<br /> <br /> * [[Geometric products]]<br /> * [[Wedge products]]</div> Eric Lengyel