Eulers Axis of Rotation Theorem states that: If A is an orthogonal 3 3 matrix for which

Chapter 7, Problem 30

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Eulers Axis of Rotation Theorem states that: If A is an orthogonal 3 3 matrix for which det(A) = 1, then multiplication by A is a rotation about a line through the origin in R3. Moreover, if u is a unit vector along this line, then Au = u. (a) Confirm that the following matrix A is orthogonal, that det(A) = 1, and that there is a unit vector u for which Au = u. A = 2 7 3 7 6 7 3 7 6 7 2 7 6 7 2 7 3 7 (b) Use Formula (3) of Section 4.9 to prove that if A is a 3 3 orthogonal matrix for which det(A) = 1, then the angle of rotation resulting from multiplication by A satisfies the equation cos = 1 2 [tr(A) 1]. Use this result to find the angle of rotation for the rotation matrix in part (a).