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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.]

2 Answers

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  • ?
    Lv 7
    9 years ago
    Favorite Answer

    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

  • 4 years ago

    Set the 2nd equation such as y. y = 2x + 5 replace that into the 2nd equation for y and simplify. x^2 + (2x + 5)^2 = 25 x^2 + 4x^2 + 20x + 25 = 25 5x^2 + 20x + 25 = 25 be sure for x. I factored after subtracting 25 from the two aspects. 5x^2 + 20x = 0 5x(x + 4) = 0 x = 0 ; x = -4 replace the two 0 and -4 for x into between the equations. it extremely is greater available to be conscious the 2nd. y - 2(0) = 5 y = 5 ... y - 2(-4) = 5 y + 8 = 5 y = -3 Your very very final ideas are... (0, 5) and (-4, -3) it rather is unquestionably a line that intersects a circle 2 circumstances :)

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