Math Problem Statement
Determine the rotation matrix for the follwing angle and direction: theta = 45^deg counter clockwise
Solution
To determine the rotation matrix for an angle in the counterclockwise direction, we use the standard 2D rotation matrix formula:
\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$$ Substituting $$\theta = 45^\circ$$: 1. $$\cos(45^\circ) = \frac{\sqrt{2}}{2}$$ 2. $$\sin(45^\circ) = \frac{\sqrt{2}}{2}$$ Thus, the rotation matrix $$R(45^\circ)$$ is: $$R(45^\circ) = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}$$ This matrix can be used to rotate a vector counterclockwise by 45 degrees. Would you like to know more details or have any further questions? Here are 5 related questions you might find useful: 1. How would the rotation matrix change for a clockwise rotation by 45 degrees? 2. What is the rotation matrix for an angle of 90 degrees? 3. How can we use this rotation matrix to rotate a vector in 3D space? 4. What is the effect of applying the rotation matrix twice in succession? 5. How does the determinant of a rotation matrix relate to its properties? **Tip:** Remember that rotation matrices are orthogonal, which means their inverse is equal to their transpose.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Trigonometry
Rotation Matrices
Formulas
2D Rotation Matrix
Theorems
-
Suitable Grade Level
Advanced High School