Algebra and Geometry | Part IA, 2002

Suppose that a,b, c,d\mathbf{c}, \mathbf{d} are the vertices of a regular tetrahedron TT in R3\mathbb{R}^{3} and that a=(1,1,1),b=(1,1,1),c=(1,1,1),d=(1,x,y)\mathbf{a}=(1,1,1), \mathbf{b}=(-1,-1,1), \mathbf{c}=(-1,1,-1), \mathbf{d}=(1, x, y).

(a) Find xx and yy.

(b) Find a matrix MM that is a rotation leaving TT invariant such that Ma=bM \mathbf{a}=\mathbf{b} and Mb=a.M \mathbf{b}=\mathbf{a} .

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