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# let T1 be the linear transformation corresponding to a rotation by an angle fo theta(1) about the x-axis in?

let T1 be the linear transformation corresponding to a rotation by an angle fo theta(1) about the x-axis in R^3, and T2 the linear transformation corresponding to a rotation by an angle of theta(2) about the z-axis.

a)Compute the standard matrices for T1 and T2

b)For what values of theta(1) and theta(2) does T2*T1=T1*T2? [Hint: you will need to consider the different possible cases, and to know a little geometry.]

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The standard rotation in xy plane is given by the matrix { {cosθ,−sinθ}, {sinθ,cosθ} }.

In 3D this is seen as a rotation about OZ with the positive sense as anti-clockwise when viewed from the +z side of the origin. The same convention is applied to rotations about OX & OY.

The 3D matrix for this rotation about OZ is { {cosθ,−sinθ,0}, {sinθ,cosθ,0}, {0,0,1} }.

For OX or OY the matrix is similar with non-zero terms placed in suitable locations.

For a rotation θ₁ about OX, T₁ = { {1,0,0}, {0,cosθ₁,−sinθ₁}, {0,sinθ₁,cosθ₁} }

For a rotation θ₂ about OZ, T₂ = { {cosθ₂,−sinθ₂,0}, {sinθ₂,cosθ₂,0}, {0,0,1} }

( although not needed, note that a rotation about OY requires a change of sign of θ )

T₁T₂ = { { c₂, −s₂, 0 }, { c₁s₂, c₁c₂, −s₁ }, { s₁s₂, s₁c₂, c₁ } }

T₂T₁ = { { c₂, −c₁s₂, s₁s₂ }, { s₂, c₁c₂, −s₁c₂ }, { 0, s₁, c₁ } }

Equating corresponding unlike terms gives (i) c₁s₂ = s₂ (ii) s₁s₂ = 0 (iii) s₁ = s₁c₂

(ii) → s₁=0 or s₂=0

If s₁=0 (iii) is satisfied and θ₁=0,π

If θ₁=0 then (i) is true for all θ₂

If θ₁=π then (i) implies s₂=0 → θ₂=0,π

If s₂=0 there are similar results

∴ T₁T₂ = T₂T₁ if either of the rotations is zero or both of them are π

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