Matrix A=[ 0 0 1; 1 0 0; 0 1 0] rotates about axis if vector B=[1 1 1] find angle of rotation?

Question

Matrix A=[ 0 0 1; 1 0 0; 0 1 0] rotates about axis if vector B=[1 1 1] find angle of rotation? Where I can find relevant information. and elementary methods to solve this kind of problem.

Answer 1

I assume the elements of the rotation matrix A are set up in terms of the standard Euler angles φ, θ, and ψ.  You can find information about Euler angles in an engineering math text book or a mathematical physics text book, under the subject of finite rotations or direction cosine matrices.  (Physics books refer to these matrices as Euler rotation matrices, while engineering books refer to them as direction cosine matrices, but they are the same thing.)  If the elements of A are in terms of the Euler angles, then you should be able to compute the 3 Euler angles from the given elements of A.  Look up the expressions for the elements of the rotation matrix.  You will find 9 expressions.  Set each one equal to the corresponding element of A, and then solve for φ, θ, and ψ.  When you evaluate inverse trigonometric functions, you will have to correctly assign quadrants to the angles.  For that reason, you may need to use more than 3 expressions, even though there are only 3 unknowns.  The fact that all the elements of the given A are either 1 or 0 simplifies the math.

 

A₃₃ = cosθ = 0 ⇒ θ = π/2 or 3π/2

If θ is the polar angle, then it must lie between 0 and π, so θ = π/2

 

A₂₃ = cosψ sinθ = 0 and sinθ ≠ 0, so cosψ = 0 ⇒ ψ = π/2 or 3π/2

 

A₁₃ = sinΨ sinθ = 1 = sinΨ sin(π/2) = sinΨ (1) ⇒ ψ = π/2

 

A₃₂ = -sinθ cosφ = -sin(π/2) cosφ = 1 = -(1)cosφ = -cosφ ⇒ cosφ = -1 ⇒ φ = π

 

It looks like the 3 Euler angles are:

φ = π

θ = π/2

ψ = π/2

 

Test this answer against the other elements of A.

 

A₁₁ = cosψ cosφ - cosθ sinφ sinψ = 0 - 0 = 0  [good]

A₂₁ = -sinψ cosφ - cosθ sinφ cosψ = -(1)(-1) - 0 = 1  [good]

A₃₁ = sinθ sinφ = 0  [good]

 

A₁₂ = cosψ sinφ + cosθ cosφ sinΨ = 0 + 0 = 0  [good]

A₂₂ = -sinψ sinφ + cosθ cosφ cosψ = 0 + 0 = 0  [good]

A₃₂ = -sinθ cosφ = -(1)(-1) = 1  [good]

 

We already used A₁₃, A₂₃, and A₃₃, so we are good.  Generally, this can be a long and tedious process, but the simple elements of A made this problem relatively easy.