CEOMETRT: Matrix multiplication can be used to rotate figures in the plane. What is the matrix product?

Certainly! Matrix multiplication is a fundamental tool in linear algebra, especially for performing transformations such as rotations in the plane. Let's delve into how matrix multiplication achieves the rotation of figures in a two-dimensional plane. ## **Rotation Matrices in 2D** A **rotation matrix** \( R(\theta) \) rotates a point in the plane by an angle \( \theta \) (in radians) about the origin. The standard rotation matrix is defined as: \[ R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix} \] ## **Matrix Multiplication for Rotation** When you want to perform two consecutive rotations—first by an angle \( \theta \) and then by an angle \( \phi \)—you can achieve this by multiplying their corresponding rotation matrices. Let's define the two rotation matrices: \[ R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix}, \quad R(\phi) = \begin{bmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \\ \end{bmatrix} \] **Multiplying \( R(\theta) \) and \( R(\phi) \):** \[ R(\theta) \cdot R(\phi) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix} \begin{bmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \\ \end{bmatrix} \] **Calculating the Product:** 1. **First Row, First Column:** \[ (\cos\theta \cdot \cos\phi) + (-\sin\theta \cdot \sin\phi) = \cos\theta\cos\phi - \sin\theta\sin\phi = \cos(\theta + \phi) \] 2. **First Row, Second Column:** \[ (\cos\theta \cdot -\sin\phi) + (-\sin\theta \cdot \cos\phi) = -\cos\theta\sin\phi - \sin\theta\cos\phi = -\sin(\theta + \phi) \] 3. **Second Row, First Column:** \[ (\sin\theta \cdot \cos\phi) + (\cos\theta \cdot \sin\phi) = \sin\theta\cos\phi + \cos\theta\sin\phi = \sin(\theta + \phi) \] 4. **Second Row, Second Column:** \[ (\sin\theta \cdot -\sin\phi) + (\cos\theta \cdot \cos\phi) = -\sin\theta\sin\phi + \cos\theta\cos\phi = \cos(\theta + \phi) \] **Putting It All Together:** \[ R(\theta) \cdot R(\phi) = \begin{bmatrix} \cos(\theta + \phi) & -\sin(\theta + \phi) \\ \sin(\theta + \phi) & \cos(\theta + \phi) \\ \end{bmatrix} = R(\theta + \phi) \] ## **Interpretation** The product \( R(\theta) \cdot R(\phi) \) results in a new rotation matrix \( R(\theta + \phi) \), which means that applying two consecutive rotations—first by \( \theta \) and then by \( \phi \)—is equivalent to a single rotation by the angle \( \theta + \phi \). ## **Example** Suppose you want to rotate a figure first by \( 30^\circ \) and then by \( 45^\circ \). 1. Convert degrees to radians: \[ 30^\circ = \frac{\pi}{6} \text{ radians}, \quad 45^\circ = \frac{\pi}{4} \text{ radians} \] 2. Multiply the rotation matrices: \[ R\left(\frac{\pi}{6}\right) \cdot R\left(\frac{\pi}{4}\right) = R\left(\frac{\pi}{6} + \frac{\pi}{4}\right) = R\left(\frac{5\pi}{12}\right) \] This demonstrates that the combined rotation is \( 75^\circ \), which is the sum of \( 30^\circ \) and \( 45^\circ \). ## **Conclusion** Matrix multiplication effectively combines multiple rotations into a single rotation by adding their respective angles. This property is highly useful in computer graphics, robotics, and various fields of engineering and physics where composite rotations are common.