Vectors and Geometry

Vectors in three dimensions form a particular subset of three-dimensional Clifford numbers. Although three-dimensional Clifford numbers have four grades and eight components, vectors retain only one of the grades and three of the components. The magnitude of a real valued vector is the square root of the sum of the squares of all of its coefficients. The signature plays no role, so the magnitude of a vector having the same components with either signature is the same. The inverse of a vector is another vector. A unit vector is used as a reference to determine what should happen when other vectors undergo a reflection. As with reflection, a unit reference vector is used to determine what happens when a vector undergoes a projection operation. Rotation is achieved by a sequence of two different reflections.