Question

What are the transition matrices for rotations through \(\pm 2 \pi / 3\) about an axis aligned with the vector with \(\mathcal{T}\)-components \((1,1,1) ?\)

Short answer

Answer: The transition matrices for the rotations are: For +2π/3 rotation: $$R\left(\frac{2\pi}{3}\right) = \begin{bmatrix} 0 & -\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\ \sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & \frac{1}{3} \end{bmatrix}.$$ For -2π/3 rotation: $$R\left(-\frac{2\pi}{3}\right) = \begin{bmatrix} 0 & \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\ -\sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & \frac{1}{3} \end{bmatrix}.$$

Step-by-step solution

Find the unit vector in the direction of the given vector

First, we need to find the unit vector of the given vector (1,1,1). To do this, we calculate the magnitude of the vector and then divide the vector by its magnitude. Magnitude of the vector: $$\lVert \mathbf{v} \rVert = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}.$$ Unit vector of the given vector: $$\mathbf{\hat{u}} = \frac{\mathbf{v}}{\lVert \mathbf{v} \rVert} = \frac{1}{\sqrt{3}}(1,1,1).$$

Rodrigues' rotation formula

Rodrigues' rotation formula is used for finding a resulting vector by rotating an initial vector by a given angle around a given axis (unit vector). The formula is as follows: $$\mathbf{v}_{rot} = \mathbf{v}\cos{\theta} + (\mathbf{\hat{u}} \times \mathbf{v})\sin{\theta} + \mathbf{\hat{u}}(\mathbf{\hat{u}} \cdot \mathbf{v})(1 - \cos{\theta}).$$

Find transition matrices for the given rotations

We can write the Rodrigues' rotation formula in matrix form: $$\mathbf{v}_{rot} = R(\theta)\mathbf{v},$$ where \(R(\theta)\) is the transition matrix. To find the transition matrices for rotations through +2π/3 and -2π/3 about the given axis, we need to substitute the corresponding angles and the unit vector obtained in step 1 into Rodrigues' rotation formula and rewrite it in matrix form. Using the unit vector \(\mathbf{\hat{u}} = \frac{1}{\sqrt{3}}(1,1,1)\) and angles \(\theta = \pm \frac{2\pi}{3}\), we substitute the values into the Rodrigues' rotation formula written in a matrix form. Rotational matrix for +2π/3 rotation: $$R\left(\frac{2\pi}{3}\right) = \begin{bmatrix} 0 & -\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\ \sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & \frac{1}{3} \end{bmatrix}.$$ Rotational matrix for -2π/3 rotation: $$R\left(-\frac{2\pi}{3}\right) = \begin{bmatrix} 0 & \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\ -\sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & \frac{1}{3} \end{bmatrix}.$$

Key concepts

Rodrigues' Rotation Formula

Understanding the Rodrigues' rotation formula is pivotal for anyone dealing with three-dimensional rotations, such as those encountered in physics, engineering, and computer graphics.

Rodrigues' formula provides a straightforward way to calculate the rotation of a vector around a specified axis by a certain angle. It is particularly valuable because it simplifies the otherwise complex trigonometric operations involved in rotations.

Let's distill the formula: given a vector \(\mathbf{v}\), an axis of rotation through a unit vector \(\mathbf{\hat{u}}\), and an angle \(\theta\), the rotated vector \(\mathbf{v}_{rot}\) is found using
\[\mathbf{v}_{rot} = \mathbf{v}\cos{\theta} + (\mathbf{\hat{u}} \times \mathbf{v})\sin{\theta} + \mathbf{\hat{u}}(\mathbf{\hat{u}} \cdot \mathbf{v})(1 - \cos{\theta}).\]
The cross product \(\mathbf{\hat{u}} \times \mathbf{v}\) accounts for the perpendicular portion of \(\mathbf{v}\) with respect to the axis, while the dot product \(\mathbf{\hat{u}} \cdot \mathbf{v}\) finds the parallel component which remains unaffected by the rotation.

When students tackle problems involving rotations like the exercise provided, visualizing the effect of the rotation significantly helps. Creating a diagram or using simulation software can further aid in comprehending the outcome of applying Rodrigues' formula to a particular scenario.

Transition Matrices

Transition matrices are fundamental tools in linear algebra, widely used to represent linear transformations, including rotations in space. In the context of analytical dynamics, a transition matrix can describe how a vector or a coordinate system changes after a rotation.

To bridge the gap between the abstract concept of rotation and its practical computation, we write Rodrigues' rotation formula as a matrix multiplication:\[\mathbf{v}_{rot} = R(\theta)\mathbf{v}.\]
A transition matrix, \(R(\theta)\), encapsulates the rotation operation and provides a convenient way to apply the same rotation to multiple vectors. Transition matrices are particularly useful in studies of rigid body motion and can be applied to problems involving changes in reference frames.

In the example solved in the exercise, we computed the transition matrices for rotations through \(\pm 2 \pi / 3\) around an axis. These matrices are built by substituting the angle \(\theta\) and unit vector \(\mathbf{\hat{u}}\) into Rodrigues' rotation formula, then arranging the coefficients in a square matrix. Having this matrix, any vector can be rotated by simply multiplying it by this matrix.

When analyzing complex systems, constructing and understanding transition matrices is a critical skill. It's advantageous to practice building these matrices from scratch and employing them in diverse rotation scenarios.

Unit Vectors

Unit vectors play an indispensable role in physics and engineering by providing direction without magnitude. These vectors have a length, or magnitude, of one, making them the perfect candidates for describing directions in space.

The adventure of the unit vector begins with any non-zero vector \(\mathbf{v}\). To convert this vector into a unit vector \(\mathbf{\hat{u}}\), one must divide \(\mathbf{v}\) by its magnitude:\[\mathbf{\hat{u}} = \frac{\mathbf{v}}{\| \mathbf{v} \|}.\]
As shown in the initial step of the exercise, the creation of a unit vector along the direction of \((1,1,1)\) is pivotal for determining the axis about which we'll rotate. This standard procedure of normalizing a vector is elementary but vital.

Students should remember that the direction represented by a unit vector remains the same regardless of the original vector's magnitude. Unit vectors are often designated with a 'hat' symbol, as in \(\mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}\), which represent the fundamental directions in Cartesian coordinates.

Grasping the concept of unit vectors is beneficial not only in problems involving rotations but also in vector projection, force resolution, and many other areas where vectors are utilized.