Geometric Algebra

In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not for all elements) and addition of objects of different dimensions.

Geometric Product

The geometric product of two vectors gives a mixed-grade object consisting of a scalar part (their dot product) and a bivector part (their wedge product).

\(\overset\rightharpoonup u\overset\rightharpoonup v=\overset\rightharpoonup u\cdot\overset\rightharpoonup v+\overset\rightharpoonup u\wedge\overset\rightharpoonup v\)

\((|A||B|)^2 = |A * B|^2 + |A \times B|^2\)

Rotations, in any dimension

We noted complex numbers excel at describing rotations in two dimensions, and quaternions in three. Geometric algebra complex numbers excel at describing rotations in any dimension.

Let \(i\) be the product of two orthonormal vectors representing the plane in which we wish to rotate. Let \(theta\) be the angle about the origin we wish to rotate. Let \(u\) be a vector.

Decompose \(u\) with respect to \(i\):

\(u = u_{\perp} + u_{\parallel}\)

That is, we find vectors \(u_{\perp} \cdot i = 0\) and \(u_{\parallel} \wedge i = 0\) satisfying the above summation. These are unique and readily computed, but for this section we only need their existence.

Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.

Clifford Algebra? and Geometric, Grassmann, Exterior Algebras Pauli Spinors Weyl Spinors