wolfe8
  • wolfe8
HALP! Determine the new components of a vector A = (1, 2, 3) after rotations through Euler angles of 30°, 60°, 30°.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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John_ES
  • John_ES
If you apply the rotation matrix for each angle, \[R=\left(\begin{matrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{matrix}\right)\]you'll have, \[R_{30}R_{60}R_{30}\vec{v}\] The result of this multiplication is the answer you asked for.
wolfe8
  • wolfe8
Wait that's it? Ohhh on it.
John_ES
  • John_ES
My soultion is, \[(-1/2-\sqrt{3},-1+\sqrt{3}/2,3)\]

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John_ES
  • John_ES
Supposing the rotation is around z axis and counterclockwise. Also, the result is the same as if you apply a rotation of 120 degrees.
wolfe8
  • wolfe8
Can you tell me what Euler angles are?
John_ES
  • John_ES
Oh, now I catch it. Sorry, it was mistake from me. It is not a rotation, those angles are the Euler angles you need. Better explained here, http://en.wikipedia.org/wiki/Euler_angles
wolfe8
  • wolfe8
How does this change what we have done?
John_ES
  • John_ES
What we have done does a rotation around the z axis first 30 degrees, then 60 degrees, and last 30 degrees more. But Eluer angles tell you a "special" rotation through three turns (the three angle they give you in the problem) but not all around the z axis.
John_ES
  • John_ES
However, you can reduce the calculus to three matrix, -Rotation around x axis (30) -Rotation around y axis (60) -Rotation around z axis (30). Multiply the matrix of each rotation and the vector, and this will give you the correct answer.
John_ES
  • John_ES
\[R_1=\left(\begin{matrix} 1 & 0 & 0 \\ 0 & \cos30 & -\sin30\\ 0 & \sin30 & \cos30\end{matrix}\right)\] \[R_2=\left(\begin{matrix} \cos60 & 0 &\sin60 \\ 0 & 1 & 0\\ -\sin60 & 0 & \cos 60\end{matrix}\right)\] \[R_3=\left(\begin{matrix} \cos30 & -\sin30 & 0 \\ \cos30 & \sin30 & 0\\ 0 & 0 & 1\end{matrix}\right)\] \[R_3R_2R_1\vec{v}\approx(3.02,2.01,0.93)\]
wolfe8
  • wolfe8
Oh thanks a bunch!

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