Paper 1, Section II, C

Algebra and Geometry | Part IA, 2007

Let M:R3R3\mathcal{M}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} be the linear map defined by

xx=ax+b(n×x)\mathbf{x} \mapsto \mathbf{x}^{\prime}=a \mathbf{x}+b(\mathbf{n} \times \mathbf{x})

where aa and bb are positive scalar constants, and n\mathbf{n} is a unit vector.

(i) By considering the effect of M\mathcal{M} on n\mathbf{n} and on a vector orthogonal to n\mathbf{n}, describe geometrically the action of M\mathcal{M}.

(ii) Express the map M\mathcal{M} as a matrix MM using suffix notation. Find a,ba, b and n\mathbf{n} in the case

M=(222221212)M=\left(\begin{array}{rrr} 2 & -2 & 2 \\ 2 & 2 & -1 \\ -2 & 1 & 2 \end{array}\right)

(iii) Find, in the general case, the inverse map (i.e. express x\mathbf{x} in terms of x\mathbf{x}^{\prime} in vector form).

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