Dot products

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The dot product is a special product in geometric algebra that makes up part of the geometric product. There are two products with symmetric properties called the dot product and antidot product.

The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ generally removes the projection of each operand onto the other, and the magnitude of the result indicates how close they are to being parallel. If the spatial extents of $$\mathbf a$$ and $$\mathbf b$$ are orthogonal, then their dot product is zero. (Compare this to the wedge product, which is zero whenever $$\mathbf a$$ and $$\mathbf b$$ are parallel.)

Scalars, regarded as multiples of the basis element $${\mathbf 1}$$, are orthogonal to everything other than scalars. The dot product between two scalars $$a$$ and $$b$$ is the ordinary product $$ab$$, but the dot product between a scalar and any other basis element is zero, as shown in the table below.

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 is commutative for operands of the same grade. In particular, we always have

$$\mathbf v \bullet \mathbf w = \mathbf w \bullet \mathbf v$$

for vectors $$\mathbf v$$ and $$\mathbf w$$. The dot product is not commutative in general, however. For general basis elements $$\mathbf a$$ and $$\mathbf b$$, reversing the order of the operands satisfies the relationship

$$\mathbf a \bullet \mathbf b = (-1)^{(\operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b))\min(\operatorname{gr}(\mathbf a), \operatorname{gr}(\mathbf b))} \mathbf b \bullet \mathbf a$$ .

The dot product subtracts the grades of its operands, so we have

$$\operatorname{gr}(\mathbf a \bullet \mathbf b) = |\operatorname{gr}(\mathbf a) - \operatorname{gr}(\mathbf b)|$$ .

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


DotProduct.svg

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 projective geometric algebra $$\mathcal G_{3,0,1}$$.


AntidotProduct.svg

De Morgan Laws

As with the other products in geometric algebra, the dot product and its dual are related by De Morgan Laws as follows.

$$\overline{\mathbf a \mathbin{\unicode{x25CF}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x25CB}} \overline{\mathbf b}$$
$$\overline{\mathbf a \mathbin{\unicode{x25CB}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x25CF}} \overline{\mathbf b}$$
$$\underline{\mathbf a \mathbin{\unicode{x25CF}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x25CB}} \underline{\mathbf b}$$
$$\underline{\mathbf a \mathbin{\unicode{x25CB}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x25CF}} \underline{\mathbf b}$$

See Also