Question

a. Find the rotation of \(\mathbf{v}=\left[\begin{array}{r}2 \\ 3 \\\ -1\end{array}\right]\) about the \(z\) axis through \(\theta=\frac{\pi}{4}\). b. Find the rotation of \(\mathbf{v}=\left[\begin{array}{l}1 \\ 0 \\\ 3\end{array}\right]\) about the \(z\) axis through \(\theta=\frac{\pi}{6}\).

Short answer

1. Part a: \( \begin{bmatrix} -\frac{1}{\sqrt{2}} \\ \frac{5}{\sqrt{2}} \\ -1 \end{bmatrix} \); Part b: \( \begin{bmatrix} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \\ 3 \end{bmatrix} \).

Step-by-step solution

Define the Rotation Matrix for the z-axis

The rotation matrix for a rotation about the z-axis by an angle \( \theta \) is \[R_{z}(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta & 0 \\sin\theta & \cos\theta & 0 \0 & 0 & 1\end{bmatrix}\]This matrix will be used for both parts of the problem with different values of \( \theta \).

Calculate Rotation for Part a

For part a, \( \theta = \frac{\pi}{4} \). Applying this angle to the rotation matrix:\[R_{z}\left(\frac{\pi}{4}\right) = \begin{bmatrix}\cos\left(\frac{\pi}{4}\right) & -\sin\left(\frac{\pi}{4}\right) & 0 \\sin\left(\frac{\pi}{4}\right) & \cos\left(\frac{\pi}{4}\right) & 0 \0 & 0 & 1\end{bmatrix} = \begin{bmatrix}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \0 & 0 & 1\end{bmatrix}\]Now multiply this matrix by the vector \( \mathbf{v} = \begin{bmatrix} 2 \ 3 \ -1 \end{bmatrix} \):\[\begin{bmatrix}2 \3 \-1\end{bmatrix} = \begin{bmatrix}\frac{\sqrt{2}}{2} \times 2 - \frac{\sqrt{2}}{2} \times 3 \\frac{\sqrt{2}}{2} \times 2 + \frac{\sqrt{2}}{2} \times 3 \-1\end{bmatrix} = \begin{bmatrix}-\frac{1}{\sqrt{2}} \\frac{5}{\sqrt{2}} \-1\end{bmatrix}\]

Calculate Rotation for Part b

For part b, \( \theta = \frac{\pi}{6} \). Applying this angle to the rotation matrix:\[R_{z}\left(\frac{\pi}{6}\right) = \begin{bmatrix}\cos\left(\frac{\pi}{6}\right) & -\sin\left(\frac{\pi}{6}\right) & 0 \\sin\left(\frac{\pi}{6}\right) & \cos\left(\frac{\pi}{6}\right) & 0 \0 & 0 & 1\end{bmatrix} = \begin{bmatrix}\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \0 & 0 & 1\end{bmatrix}\]Now multiply this matrix by the vector \( \mathbf{v} = \begin{bmatrix} 1 \ 0 \ 3 \end{bmatrix} \):\[\begin{bmatrix}\frac{\sqrt{3}}{2} \times 1 - \frac{1}{2} \times 0 \\frac{1}{2} \times 1 + \frac{\sqrt{3}}{2} \times 0 \3\end{bmatrix} = \begin{bmatrix}\frac{\sqrt{3}}{2} \\frac{1}{2} \3\end{bmatrix}\]

Short Answer

The rotated vector for part a is \( \begin{bmatrix} -\frac{1}{\sqrt{2}} \ \frac{5}{\sqrt{2}} \ -1 \end{bmatrix} \) and for part b is \( \begin{bmatrix} \frac{\sqrt{3}}{2} \ \frac{1}{2} \ 3 \end{bmatrix} \).

Key concepts

Rotation Matrix

Understanding the rotation matrix is essential when working with 3D vector rotations. A rotation matrix is used to perform a rotation in a coordinate space. It acts as a linear transformation, allowing you to change the orientation of objects in a 3D environment.

The rotation matrix for a rotation about the z-axis, as we see in our exercise, involves trigonometric functions such as cosine and sine. It takes the form:

• \[R_{z}( heta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \ \sin(\theta) & \cos(\theta) & 0 \ 0 & 0 & 1 \end{bmatrix}\]

This matrix will rotate a vector around the z-axis by an angle \(\theta\). Notice that the z-component of the vector remains unchanged since the rotation is only happening on the xy-plane. To apply the rotation, multiply this matrix with the vector you want to rotate.

Z-axis Rotation

Z-axis rotation is a specific type of 3D rotation where the object rotates around the z-axis. Imagine a spinning record on a turntable—this is similar to what a z-axis rotation does to a vector.

The z-axis does not move, but the vector's x and y coordinates change as they are rotated. It’s important because many practical applications, such as camera movements in animations or rotations in robotics, utilize z-axis rotations. Particularly, our exercise explores rotating two vectors by different angles about the z-axis, first by \(\frac{\pi}{4}\) and then \(\frac{\pi}{6}\).

Z-axis rotations are common in computer graphics and game development where the orientation of items often needs to be adjusted without altering their vertical position.

Trigonometric Functions

Trigonometric functions, especially sine and cosine, play a crucial role in rotations involving angles. They are used to calculate the new positions of the points during a rotation.

For instance, rotating around the z-axis involves these trigonometric functions to determine the new x and y coordinates of a vector. The cosine function helps to find the new projection along the original x-axis, while the sine function adjusts the projection according to the angle of rotation.

In our exercise, cosine and sine values of \(\frac{\pi}{4}\) resulted in \(\frac{\sqrt{2}}{2}\), while for \(\frac{\pi}{6}\), they were \(\frac{\sqrt{3}}{2}\) and \(\frac{1}{2}\). Trigonometry thus provides a mathematical way to measure how much we move along each axis from the vector's initial position during rotation.

Linear Algebra

Linear algebra underpins many calculations involving matrices and vectors, like those in our exercise. It allows us to transform objects, rotate them, and even apply scaling.

Multiplying a matrix by a vector, as shown in our solution, is a linear operation. It involves combining rows of the matrix with columns of the vector to get results in a new vector. This operation uses the rules of dot product, leading to the rotated coordinates.

Linear algebra is a powerful tool because it provides the framework to do more complex transformations, like sequential rotations or translations, by simply multiplying different matrices together. It’s widely used in engineering, physics, computer graphics, and computer vision for its properties of linearity and ease of computation.