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This problem involves determining a rotation matrix based on two sequential rotations applied to a coordinate frame. Here's the step-by-step approach:

1. **Initial Condition**: The frames \( \{A\} \) and \( \{B\} \) are initially coincident, meaning their coordinate axes are aligned.

2. **First Rotation**: The frame \( \{B\} \) is rotated about its \( \hat{Z}_B \)-axis by an angle \( \theta \).

3. **Second Rotation**: After the first rotation, the new orientation of frame \( \{B\} \) is rotated about its \( \hat{X}_B \)-axis by an angle \( \phi \).

4. **Objective**: To determine the rotation matrix that transforms a vector \( \mathbf{P} \) described in frame \( B \), denoted as \( B\mathbf{P} \), to the same vector described in frame \( A \), denoted as \( A\mathbf{P} \).

### Steps to Solve:

1. **Rotation about \( \hat{Z}_B \)**: A rotation about the \( \hat{Z} \)-axis by an angle \( \theta \) is represented by the following rotation matrix:
   \[
   R_Z(\theta) = \begin{pmatrix}
   \cos\theta & -\sin\theta & 0 \\
   \sin\theta & \cos\theta & 0 \\
   0 & 0 & 1
   \end{pmatrix}
   \]

2. **Rotation about \( \hat{X}_B \)**: After the rotation by \( \theta \), the frame is rotated about the new \( \hat{X}_B \)-axis by an angle \( \phi \). A rotation about the \( \hat{X} \)-axis by \( \phi \) is represented by:
   \[
   R_X(\phi) = \begin{pmatrix}
   1 & 0 & 0 \\
   0 & \cos\phi & -\sin\phi \\
   0 & \sin\phi & \cos\phi
   \end{pmatrix}
   \]

3. **Combined Rotation**: The combined rotation matrix that transforms from \( B \) to \( A \) is the product of the two rotations:
   \[
   R = R_X(\phi) R_Z(\theta)
   \]
   where \( R_Z(\theta) \) is applied first, followed by \( R_X(\phi) \).

Let's compute the combined matrix \( R \).

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

Multiplying these matrices:

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

### Final Answer:

The rotation matrix that transforms vectors from frame \( B \) to frame \( A \) is:
\[
R = \begin{pmatrix}
   \cos\theta & -\sin\theta & 0 \\
   \cos\phi\sin\theta & \cos\phi\cos\theta & -\sin\phi \\
   \sin\phi\sin\theta & \sin\phi\cos\theta & \cos\phi
   \end{pmatrix}
\]

Let me know if you'd like further details or a more in-depth explanation.

### Related Questions:
1. How do you derive the rotation matrix for a rotation about the \( Y \)-axis?
2. What is the inverse of the combined rotation matrix \( R \)?
3. How does the rotation matrix change if the rotations are applied in a different order?
4. Can you compute the determinant of a general rotation matrix?
5. What are the properties of an orthogonal matrix, and why are rotation matrices orthogonal?

**Tip**: Always remember that the order of applying rotations matters in three-dimensional space due to the non-commutative nature of matrix multiplication.

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 a frame {B} that undergoes sequential rotations, first about the Z-axis by θ degrees and then about the X-axis by φ degrees. This problem covers core concepts of linear algebra and rotation matrices.

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