Have been revising geometry today and have came across some proofs that I can't seem to find in books, but I can't get through either. Any help would be great.(adsbygoogle = window.adsbygoogle || []).push({});

Let A be a 3x3 orthogonal matrix and let x and y be vectors in R^3

a) Show that detA = +/- 1

b) Show that the length of Ax is the same as that of x and that x and y are orthogonal iff Ax and Ay are orthogonal

Suppose further that A represents a rotation through angleθ , with axis of rotation along the unit vector n, show that if m is a unit vector orthogonal to n, then n.m^Am = sinθ

attempt at a)

from defn of orthogonal matrix (Atr.A = I)

det(Atr . A) = det(I) = 1

using standard results such as det(A.B) = detA. detB and det(Atr) = det(A)

we have det(Atr).det(A) = det(A)^2

detA^2=1

therefore det A = +\- 1

b) struggling to start.

Thank you in advance

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# Isometries Proofs

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