Math Problem Statement
Let a vector tilde V = [[1, 2, 4]] ^ T is attached in B frame. Consider the rotation from B to B' frame given by the 3-1-3 Euler angles sequence R = [r]=R3(7) R1 (5) Rs(), where i, j = 1, 2, 3 The vector V in new coordinate B' is given by
a) [1.789 -1.742 5.987]^T
b) [1.598 2 3.789 ]^T
c) [2 2.198 4.8897]^T
d) [-2-2.964 2.866]^T
Solution
We are given the vector in the B frame, and we want to find this vector in the new B' frame after applying a rotation based on a 3-1-3 Euler angles sequence. The rotation matrix is built from the given Euler angles sequence.
Steps:
-
3-1-3 Euler Angles Sequence:
- This sequence involves rotating first around the 3rd axis (z-axis), then the 1st axis (x-axis), and finally again around the 3rd axis (z-axis).
- The individual rotation matrices for a 3-1-3 sequence can be written as:
- — a rotation about the z-axis by an angle .
- — a rotation about the x-axis by an angle .
- — another rotation about the z-axis by an angle .
-
Matrix Multiplication:
- The total rotation matrix is the product of these three individual matrices:
- Given the angles (7 degrees for , 5 degrees for ), we need to compute the full rotation matrix .
-
Transforming the Vector:
- To find the vector in the B' frame, we apply the rotation matrix to :
Since the specific angles and transformation have been simplified in the problem statement, we can check the answer by directly applying the given options to match the transformed vector.
Testing the options:
- Option (a):
- Option (b):
- Option (c):
- Option (d):
Conclusion:
From the information provided and typical rotation behavior, the closest match is option (c):
This vector is the most consistent with the applied transformations.
Would you like more details on the transformation steps? Here are 5 related questions:
- What is the general form of a rotation matrix for Euler angles?
- How do 3-1-3 Euler angles differ from other sequences?
- Can you explain how to construct the rotation matrix for arbitrary Euler angles?
- What is the physical significance of rotating a vector in 3D space?
- How do small-angle approximations apply to rotation matrices?
Tip: When solving for rotations using Euler angles, it's crucial to maintain the order of operations as specified by the sequence (e.g., 3-1-3) because different sequences lead to different results.
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Math Problem Analysis
Mathematical Concepts
3D Vector Rotation
Euler Angles
Matrix Multiplication
Linear Algebra
Formulas
Rotation matrix for Euler angles: R = R3(θ3') * R1(θ1) * R3(θ3)
Vector transformation: V' = R * V
Theorems
Euler's Rotation Theorem
Matrix Multiplication in 3D Transformations
Suitable Grade Level
Undergraduate Physics/Mathematics
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