Math Problem Statement
Can you give me the answer of this matrix?
Can you give me the answer of this matrix?
Solution
The matrix you have shared is a transformation matrix often used in the context of 3D rotations or Lorentz transformations. The matrix is:
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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
3D Rotation Matrices
Trigonometry
Formulas
[L]^u_u = [[1, 0, 0], [0, cos(φ), -sin(φ)], [0, sin(φ), cos(φ)]]
Theorems
Rotation Matrix
Orthogonal Matrices
Suitable Grade Level
Undergraduate Engineering or Physics
Related Recommendation
Rotation of a Vector about X and Y Axes to Coincide with Z-Axis
Simplify Matrix Expression Involving Rotation Matrices and Vector Operations
Understanding the Angle of Rotation in Rotating Bodies
Rotation Matrix for Sequential Rotations in 3D Space: θ and φ
Understanding Rotations: Applications in Math, Physics, and Computer Graphics