Question

(a) Find the 3 \(\times 3\) matrix \(\mathbf{R}(\theta)\) that rotates three- dimensional space about the \(x_{3}\) axis, so that \(\mathbf{e}_{1}\) rotates through angle \(\theta\) toward \(\mathbf{e}_{2}\). (b) Show that \([\mathbf{R}(\theta)]^{2}=\mathbf{R}(2 \theta),\) and interpret this result.

Short answer

Matrix \(\mathbf{R}(\theta)\) represents rotation, and \([\mathbf{R}(\theta)]^2 = \mathbf{R}(2\theta)\) illustrates additive rotation.

Step-by-step solution

Define the Rotation Matrix

The rotation matrix that rotates about the \(x_3\) axis is of the form: \[ \mathbf{R}(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} \] This matrix keeps the \(x_3\) component unchanged while rotating the \(x_1\) and \(x_2\) components.

Verify the Rotation Matrix Properties

The matrix \(\mathbf{R}(\theta)\) is orthogonal, meaning \(\mathbf{R}(\theta)^{T} \mathbf{R}(\theta) = \mathbf{I}\), where \(\mathbf{I}\) is the identity matrix. Each column of \(\mathbf{R}(\theta)\) has a magnitude of 1, and the columns are mutually orthogonal.

Compute \( [\mathbf{R}(\theta)]^2 \)

Calculate \([\mathbf{R}(\theta)]^2 = \mathbf{R}(\theta) \mathbf{R}(\theta) \) by matrix multiplication: \[ \begin{bmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \cos 2\theta & -\sin 2\theta & 0 \ \sin 2\theta & \cos 2\theta & 0 \ 0 & 0 & 1 \end{bmatrix} \] This confirms that \([\mathbf{R}(\theta)]^2 = \mathbf{R}(2\theta)\).

Interpret the Result

The equation \([\mathbf{R}(\theta)]^2 = \mathbf{R}(2\theta)\) means that applying the rotation twice by angle \(\theta\) is equivalent to a single rotation by the angle \(2\theta\). This shows the rotational property of these matrices, where combining them results in additive angles.

Key concepts

Orthogonal Matrix

An orthogonal matrix is a special kind of square matrix with the property that its rows and columns are orthonormal. This means that they are all vectors with a length (or magnitude) of 1, and they are perpendicular to each other.
This unique property ensures that when an orthogonal matrix is used in transformation, it preserves the original shape and size of objects, making it very useful in 3D rotations.

• Mathematically, a matrix \( extbf{A} \) is orthogonal if \( extbf{A}^T \textbf{A} = extbf{I} \), where \( extbf{I} \) is the identity matrix.
• The transpose of \( extbf{A} \) (\( extbf{A}^T \)) is simply the matrix with its rows and columns swapped.

This property makes computations efficient and reliable, as no distortions occur during transformations.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra used to combine two matrices. It is applicable in various fields, including computer graphics, physics, and engineering.
The process involves taking the rows of the first matrix and the columns of the second matrix, multiplying corresponding entries, and summing them up.

• If \( extbf{A} \) is a \( 3 \times 3 \) matrix and \( extbf{B} \) is a \( 3 \times 3 \) matrix, their product \( extbf{C} = \textbf{A} \textbf{B} \) will also be a \( 3 \times 3 \) matrix.
• For example, to compute \( [ extbf{C}]_{1,1} \), multiply the first row of \( extbf{A} \) with the first column of \( extbf{B} \) and sum the products.

This operation is crucial in performing transformations, such as combining rotations. It allows for compositional transformations, like multiplying a rotation matrix by itself to achieve a rotation by double the angle.

Three-Dimensional Rotation

Three-dimensional rotation involves turning an object around an axis in 3D space. Imagine spinning a globe around its axis; that's a real-world example of 3D rotation.
In mathematics, this is accomplished using rotation matrices, which provide a way to calculate the new position of a point after rotation.

• For rotations about the \( x_3 \) axis, a typical rotation matrix is \( \begin{bmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} \).
• This matrix keeps the third coordinate unchanged, while the first two coordinates rotate by an angle \( \theta \).

Each component of a point in space transforms according to these rules, maintaining its relative position to the origin. Understanding these transformations helps in visualizing and handling complex 3D graphics.