https://file.aiquickdraw.com/mathbot/file/172805227381608gj304u.jpg

Can you give me the answer of this matrix?

The matrix you have shared is a transformation matrix often used in the context of 3D rotations or Lorentz transformations. The matrix is:

\[
[L]^u_u = \begin{pmatrix}
1 & 0 & 0 \\
0 & \cos\phi & -\sin\phi \\
0 & \sin\phi & \cos\phi
\end{pmatrix}
\]

### Interpretation:

1. **Type**: This is a 3x3 matrix that represents a rotation about the \(x\)-axis by an angle \(\phi\) (in radians).
   
2. **First Row**: The first row \((1, 0, 0)\) indicates that the \(x\)-coordinate remains unchanged during the transformation.

3. **Second and Third Rows**: The second and third rows contain trigonometric functions of the angle \(\phi\), which represent the standard 2D rotation matrix for the \(y\)- and \(z\)-coordinates. Specifically:
   - The \(y\)-coordinate gets transformed as:
     \[
     y' = \cos\phi \cdot y - \sin\phi \cdot z
     \]
   - The \(z\)-coordinate gets transformed as:
     \[
     z' = \sin\phi \cdot y + \cos\phi \cdot z
     \]

### Use Case:
This matrix is typically used in physics and engineering for:
- Rotations about the \(x\)-axis.
- Rotational dynamics in mechanics or quantum mechanics.
- Lorentz transformations in relativity (when considering boosts along one axis).

Would you like to delve further into the applications of this matrix, such as solving a specific problem, or need more details?

---

Here are some related questions you might find useful:

1. How do you derive the rotation matrix for different axes (e.g., \(y\)-axis or \(z\)-axis)?
2. What is the general form of a 3D rotation matrix?
3. Can we express this transformation matrix in terms of Euler angles?
4. How is this matrix applied to a vector in 3D space?
5. What are some real-world applications of such rotation matrices?

**Tip**: Remember that rotation matrices are orthogonal, meaning their transpose is equal to their inverse. This property is useful for reversing transformations.

This problem covers the concept of 3D rotation matrices, specifically about the x-axis, using trigonometric functions of the angle φ. Learn how this transformation works and how to apply it in physics and engineering contexts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation