https://file.aiquickdraw.com/mathbot/file/1727493404564384g2o4b.jpg

To address the problem in the image, the goal is to determine the rotation matrix that transforms the coordinates of a vector described in frame \( B \) to those in frame \( A \). Let's break this down step by step based on the given information:

### Problem Breakdown:

- Initially, frames \( B \) and \( A \) are coincident, meaning that they share the same orientation.
- Frame \( B \) is rotated about its \( Z_B \)-axis by an angle \( \theta \).
- After this, the new frame (still denoted as \( B \)) is further rotated about its new \( X_B \)-axis by an angle \( \phi \).

The problem asks to provide the rotation matrix that converts the description of vectors from frame \( B \) to frame \( A \), i.e., the matrix that maps the vector \( B_P \) in frame \( B \) to \( A_P \) in frame \( A \).

### Solution:

1. **First Rotation (about \( Z_B \)):**
   The first rotation is around the \( Z_B \)-axis by an angle \( \theta \). The rotation matrix for a rotation about the \( Z \)-axis is given by:

   \[
   R_z(\theta) = \begin{pmatrix}
   \cos \theta & -\sin \theta & 0 \\
   \sin \theta & \cos \theta & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \]

2. **Second Rotation (about \( X_B \)):**
   After the first rotation, the next rotation is about the \( X_B \)-axis by an angle \( \phi \). The rotation matrix for a rotation around the \( X \)-axis is:

   \[
   R_x(\phi) = \begin{pmatrix}
   1 & 0 & 0 \\
   0 & \cos \phi & -\sin \phi \\
   0 & \sin \phi & \cos \phi
   \end{pmatrix}
   \]

3. **Composite Rotation:**
   Since the first rotation is around \( Z_B \) and the second is around \( X_B \), the total rotation matrix is the product of these two individual rotation matrices:

   \[
   R = R_x(\phi) R_z(\theta)
   \]

   Substituting the expressions for \( R_z(\theta) \) and \( R_x(\phi) \), we get the total rotation matrix \( R \):

   \[
   R = \begin{pmatrix}
   1 & 0 & 0 \\
   0 & \cos \phi & -\sin \phi \\
   0 & \sin \phi & \cos \phi
   \end{pmatrix}
   \begin{pmatrix}
   \cos \theta & -\sin \theta & 0 \\
   \sin \theta & \cos \theta & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \]

4. **Final Matrix Multiplication:**
   Performing the matrix multiplication:

   \[
   R = \begin{pmatrix}
   \cos \theta & -\sin \theta & 0 \\
   \sin \theta & \cos \theta & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \begin{pmatrix}
   1 & 0 & 0 \\
   0 & \cos \phi & -\sin \phi \\
   0 & \sin \phi & \cos \phi
   \end{pmatrix}
   \]

   Performing the multiplication gives the composite rotation matrix.

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Let me know if you'd like a detailed calculation of the final matrix, or if you'd like further explanation of these concepts.

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Here are 5 follow-up questions that might help deepen your understanding:

1. How does the order of rotations affect the final rotation matrix?
2. What happens if the rotation axes are different between the two frames?
3. Can you derive the rotation matrix for any arbitrary axis?
4. How would the solution change if we had three successive rotations?
5. What is the significance of using rotation matrices in 3D transformations?

**Tip:** When multiplying rotation matrices, the order in which you multiply them is critical because matrix multiplication is not commutative.

A frame {B} is located initially coincident with a frame {A}. We rotate {B} about Ẑ_B by θ degrees, then we rotate the resulting frame about X̂_B by ϕ degrees. Give the rotation matrix that will change the descriptions of vectors from B_P to A_P.

Learn how to compute the rotation matrix for transforming vectors between two coordinate frames, B and A, after two successive rotations. The problem involves rotations about the Z and X axes and explores composite rotation matrices. Suitable for advanced high school and undergraduate students studying 3D transformations.

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