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19 Feb 2015: which can be re-arranged to give. rλ = αψ. [a (sin (λψ β1) sin β1) b (cos (λψ β1) cos β1)] λt. = ... ψ[a sin β1 b cos β1] λt. noting that in the case ψ 0, the expression becomes rλ = λt as one would ex-pect.

19 Feb 2015: R(λ) = R0(cos λθ sin λθ B̂. ), (3.22). where we have used B = θB̂. ... But we know that. exp(B) = cos θ sin θ B̂ = R̃0R1.

2 Feb 2015: The inner product ab, also known as the dot or scalar product, of two vectorsa and b, is a scalar with magnitude |a||b|cos θ, where |a| and |b| are ... Here |a| (aa) 12 , sothat the expression for ab is effectively an algebraic definition of cos θ.

19 Feb 2015: 2.11) (ba) x(ba)1. = baxa1b1 =baxab. ‖a‖2‖b‖2Let u a/‖a‖, v b/‖b‖ and R vu be the corresponding unit quaternion.Then R = cos(θ/2) ... sin( θ2 )eıφ/2. . . . . cos( ϑ2 )eıϕ/2. sin( ϑ2 )eıϕ/2.

14 Feb 2015: This can be verified fromthe fact that. eIθσ1 = (cos θ sin θI)σ1 = cos θ σ1 sin θσ2 (1.54). ... andeIθσ2 = cos θ σ2 sin θσ1. (1.55). Viewed as even elements in the 2-dimensional geometric algebra, exponentials of“imaginaries” generate