Algebra and Geometry | Part IA, 2004

State the conditions on a matrix AA that ensure it represents a rotation of R3\mathbb{R}^{3} with respect to the standard basis.

Check that the matrix

A=13(122221212)A=\frac{1}{3}\left(\begin{array}{ccc} -1 & 2 & -2 \\ 2 & 2 & 1 \\ 2 & -1 & -2 \end{array}\right)

represents a rotation. Find its axis of rotation n\mathbf{n}.

Let Π\Pi be the plane perpendicular to the axis n\mathbf{n}. The matrix AA induces a rotation of Π\Pi by an angle θ\theta. Find cosθ\cos \theta.

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