vector_rules

Algebraic identities for vector/matrix geometry operations.

Rules applied

• cross(x, x) = 0 self-cross is zero vector
• cross(x, 0) = cross(0, x) = 0 zero absorption
• dot(x, x) = 1 when x has unit norm
• dot(x, 0) = dot(0, x) = 0 zero absorption
• dot(x, cross(x, y)) = 0 orthogonality
• dot(x, cross(y, x)) = 0 orthogonality
• dot(x, cross(y, y)) = 0 self-cross is zero
• Same rules when cross is on the left side of dot (commutativity)
• dot(ω, q) = 0 tangent vector orthogonal to S2 manifold point
• cross(q, cross(ω, q)) = ω S2 tangent recovery (unit norm + orthogonality)
• Transpose(Transpose(x)) = x double transpose cancellation
• Hat(0) = ZeroMatrix hat of zero vector
• Linearity through Add and Mul