By Miller

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We also see that by using reversion the order of the factors in the quaternionic product is reversed, thus it is in fact an antiautomorphism. It is exactly this last property that justiﬁes the name reversion. 19 (Rodrigues, Porteous). An arbitrary automorphism or antiautomorphism m of the algebra H has always the representation m(x) := Sc (x) + h Vec (x) , x ∈ H, with an orthogonal automorphism h of R3 . Proof. Let m(1) = y0 + y with y0 ∈ R, y ∈ R3 . Then since m(1) = m(12 ) = m2 (1) we have y02 − |y|2 + 2y0 y = y0 + y.

39. Since | cos(x, y)| ≤ 1 we have |x · y| ≤ |x||y|; which is the well-known Schwarz’s inequality Hermann A. Schwarz (1843–1921), German mathematician, active in Halle, Zürich, Göttingen and Berlin. He worked in analysis and published important papers in function theory. We now turn to the simple geometric ﬁgures in R3 , straight lines, planes and spheres. 40 (Equation of a plane). Let n = 0 be a given vector and d a real number. 1) where y represents an arbitrary vector, orthogonal to n. 1) deﬁnes the plane through the point nd/|n|2 orthogonal to n.

I) The proofs for scalar and vector product are completely parallel. Since real numbers can be exchanged with quaternions, from r r(x · y) = − (xy + yx) 2 it follows immediately that r can be pulled inside the product near x as well as near y. (ii) Again the proof can be done in parallel for the scalar and vector products. Let us choose one of the four equations. From the distributivity of the quaternion multiplication it follows that x × (y + z) = = 1 (x(y + z) − (y + z)x) 2 1 (xy + xz − yx − zx) = x × y + x × z.