Question

Consider the matrix representations for two-dimensional rotations of vectors by angles \(\alpha\) and \(\beta\), denoted by \(R_{a}\) and \(R_{\beta}\), respectively. a. Find \(R_{a}^{-1}\) and \(R_{\alpha}^{T}\). How do they relate? b. Prove that \(R_{\alpha+\beta}=R_{\alpha} R_{\beta}=R_{\beta} R_{\alpha}\).

Short answer

The inverse of rotation matrix \(R_{\alpha}\) is equal to its transpose. Moreover, the rotation matrix corresponding to the sum of angles \(\alpha\) and \(\beta\) is the same as the matrix product of the rotation matrices for \(\alpha\) and \(\beta\), in any order.

Step-by-step solution

Define the Matrix

Let's define the rotation matrix for an angle \(\alpha\). It is given by:\[R_{\alpha}=\begin{bmatrix}\cos(\alpha) & -\sin(\alpha)\\sin(\alpha) & \cos(\alpha)\end{bmatrix}\]

Inverse of the Rotation Matrix

The inverse of a matrix is defined as the matrix that, when multiplied with the original matrix, yields the identity matrix. For the rotation matrix, that inverse is given by its transposition: \[R_{\alpha}^{-1} = R_{\alpha}^{T}\]The transpose of a matrix is found by interchanging its rows and columns. So, \[R_{\alpha}^{T} =\begin{bmatrix}\cos(\alpha) & \sin(\alpha)\-\sin(\alpha) & \cos(\alpha)\end{bmatrix}\]

Proving the Matrix Transformation

Given \(R_{\alpha + \beta}\), \(R_{\alpha}\), and \(R_{\beta}\), we need to prove that \(R_{\alpha + \beta} = R_{\alpha}R_{\beta} = R_{\beta}R_{\alpha}\). Let's denote the rotation matrixes as follows:\[R_{\alpha}=\begin{bmatrix}\cos(\alpha) & -\sin(\alpha)\\sin(\alpha) & \cos(\alpha)\end{bmatrix}\]and\[R_{\beta}=\begin{bmatrix}\cos(\beta) & -\sin(\beta)\\sin(\beta) & \cos(\beta)\end{bmatrix}\]Then, \(R_{\alpha}R_{\beta}\) and \(R_{\beta}R_{\alpha}\) give\[ \begin{bmatrix}\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) & -\cos(\alpha)\sin(\beta) -\cos(\beta)\sin(\alpha)\\cos(\beta)\sin(\alpha) + \cos(\alpha)\sin(\beta) & \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\end{bmatrix}\]This matrix is equivalent to \(R_{\alpha + \beta}\), so the equality is proven. Note: we used trigonometric identities to simplify terms.

Key concepts

Rotation Matrix

The concept of a rotation matrix in geometry is paramount for transforming vectors. It describes how a vector in a two-dimensional plane can be rotated around the origin. A typical two-dimensional rotation matrix for an angle \(\alpha\) is expressed as:

\[R_{\alpha} =\begin{bmatrix}\cos(\alpha) & -\sin(\alpha)\sin(\alpha) & \cos(\alpha)\end{bmatrix}\]
This matrix, when applied to a vector, rotates it by an angle \(\alpha\) in a counterclockwise direction. Understanding the rotation matrix is essential for many applications in computer graphics, robotics, and physics, as it helps in simulating the behavior of objects when they move through space.

Inverse Matrix

An inverse matrix is akin to the reverse gear in a car; it undoes what has been done. In the context of rotation matrices, the inverse matrix \(R_{\alpha}^{-1}\) is used to rotate a vector back by an angle \(\alpha\) after it has been rotated. Mathematically, multiplying a matrix by its inverse yields the identity matrix, effectively resetting the transformation:

\[R_{\alpha}R_{\alpha}^{-1} = I\]
For a rotation matrix, the inverse can be strategically acquired by transposing the original matrix, owing to the orthogonality of the rotation matrix.

Matrix Transposition

Matrix transposition is a method of flipping a matrix over its diagonal; row elements become column elements, and vice versa. For rotation matrices, which are symmetrical, transposition involves changing the signs of the off-diagonal elements:

\[R_{\alpha}^{T} =\begin{bmatrix}\cos(\alpha) & \sin(\alpha)\-\sin(\alpha) & \cos(\alpha)\end{bmatrix}\]
The transpose of a rotation matrix serves a dual purpose; it's not only its inverse but also represents a rotation in the opposite direction.

Matrix Multiplication

Matrix multiplication is not as straightforward as multiplying individual elements; it's more about combining the effects of two transformations. When rotating vectors in a plane, combining two rotation matrices \(R_{\alpha}\) and \(R_{\beta}\) results in a single matrix that encapsulates the effect of rotating by both angles consecutively:

\[R_{\alpha}R_{\beta} = R_{\alpha + \beta}\]
Here, the right-hand side denotes a single rotation by \(\alpha + \beta\), exhibiting the cumulative effect. This property of rotation matrices being combined through multiplication is crucial for complex transformations where multiple rotations are involved.

Trigonometric Identities in Matrices

Trigonometric identities simplify the complexities of matrix multiplication when dealing with rotation matrices. Identities like \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\) and \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\) help in reducing the components of the product of two rotation matrices to the form of a single rotation matrix, which explains why \(R_{\alpha}R_{\beta}\) is equal to \(R_{\alpha + \beta}\).

Employing these identities ensures that mathematical computations are consistent with geometric intuition. Without these identities, one would have to manually execute rotations to determine the result, which is both tedious and impractical, especially for complex rotations.