Note that if n is not a unit vector then
u = (n•n)v − 2(v•n)n
is the reflection of v multiplied by the square of the length of n.
ui = Oijvj where Oij = (∑knk2)δij − 2 ninj.
We note that the function of three vectors f(u, v; w) = (u•v)w − 2(w•u)v is linear in each of its three arguments and in particular for fixed u and v, F(u, v) = λw.f(u, v; w) is a linear operator from ℝk to ℝk. F is thus bilinear and defines the ‘quadratic form’ R(v) = F(v, v). I put “quadratic form” in quotes because quadratic forms are supposed to return scalars. If n is a unit vector then F(n, n) is the reflection about the space normal to n. I suspect this generalizes to higher rank Clifford numbers and there may be a bilinear formula for the transformation matrix in terms of Clifford numbers.
I assume that n, as an element of Ck, represents this reflection. The only other Clifford numbers that represents the same reflection in this manner is −n. I pursue the bilinear scheme here.
The Clifford number mn thus represents this rotation. If m' and n' are two other vectors in the 2-space of m and n then m'∧n' is a scalar multiple of m∧n but the real parts may differ. If the real parts differ then different orientations are denoted but if they are the same then mn = m'n' and the same Clifford number denotes the same orientation, naturally. I suppose that if m and n are unit vectors then mn is a unit bi-vector but I have not found a definition of magnitude for anything but ordinary vectors yet. Add the following code to these tools:
(define (mag x)(if (pair? x)(+ (mag (car x))(mag (cdr x))) (* x x))) (define (cis th m n)(H+ (H* (Hrls (cos th)) m) (H* (Hrls (sin th)) n))) (define pi 3.141592653589793238) (define sx (/ pi 6)) (define cr (H* (cis sx g0 g1) (cis sx g2 g3))) (mag g0) ; => 1 (mag (cis sx g0 g1)) ; => 1.0 (mag cr) ; => 1.0000000000000002This suggests that the ‘magnitude’ of the product of two ‘unit’ Clifford numbers is one despite the lack of theory for this definition of magnitude. It also suggests there is a subset of Clifford numbers closed under multiplication for which the magnitude of a product is the produce of magnitudes. But
(let ((a (even (Hsg)))(b (even (Hsg)))) (- (* (Hmag a)(Hmag b)) (Hmag (H* a b))))shows that not all even values are in this subset.
The orthogonal matrix for the rotation can be computed from the components of m and n by multiplying the matrices, as defined above, of the reflections for m and n.
It seems clear how to generalize to other multi-vectors (Clifford numbers) when the multi-vector is expressed as the product of simple vectors. I do not yet know how to find such factors of a general multi-vector. It is probably as hard as finding eigenvalues of matrices, for such factoring can be used to that end.