Algebra and Geometry | Part IA, 2006

Given a vector x=(x1,x2)R2\mathbf{x}=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}, write down the vector x\mathbf{x}^{\prime} obtained by rotating x\mathbf{x} through an angle θ\theta.

Given a unit vector nR3\mathbf{n} \in \mathbb{R}^{3}, any vector xR3\mathbf{x} \in \mathbb{R}^{3} may be written as x=x+x\mathbf{x}=\mathbf{x}_{\|}+\mathbf{x}_{\perp} where x\mathbf{x}_{\|}is parallel to n\mathbf{n} and x\mathbf{x}_{\perp} is perpendicular to n\mathbf{n}. Write down explicit formulae for x\mathbf{x}_{\|}and x\mathbf{x}_{\perp}, in terms of n\mathbf{n} and x\mathbf{x}. Hence, or otherwise, show that the linear map

xx=(xn)n+cosθ(x(xn)n)+sinθ(n×x)\mathbf{x} \mapsto \mathbf{x}^{\prime}=(\mathbf{x} \cdot \mathbf{n}) \mathbf{n}+\cos \theta(\mathbf{x}-(\mathbf{x} \cdot \mathbf{n}) \mathbf{n})+\sin \theta(\mathbf{n} \times \mathbf{x})

describes a rotation about n\mathbf{n} through an angle θ\theta, in the positive sense defined by the right hand rule.

Write equation ()(*) in matrix form, xi=Rijxjx_{i}^{\prime}=R_{i j} x_{j}. Show that the trace Rii=1+2cosθR_{i i}=1+2 \cos \theta.

Given the rotation matrix

R=12(1+r1r11r1+r1112r)R=\frac{1}{2}\left(\begin{array}{ccc} 1+r & 1-r & 1 \\ 1-r & 1+r & -1 \\ -1 & 1 & 2 r \end{array}\right)

where r=1/2r=1 / \sqrt{2}, find the two pairs (θ,n)(\theta, \mathbf{n}), with πθ<π-\pi \leqslant \theta<\pi, giving rise to RR. Explain why both represent the same rotation.

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