I'm not sure how to do the counter clockwise rotations from the origin in this example? **Topic: Linear Transformations in Euclidean Space**### Problem Statement:1.) Find the standard matrix for a counterclockwise rotation of \(210^\circ\) about the origin in \(\mathbb{R}^2\). Then use this to determine where the point \((5, 8)\) is mapped under this transformation.**Explanation:**To solve this problem, we need to construct the rotation matrix for a counterclockwise rotation by an angle \(\theta\) in the 2D plane. The standard rotation matrix for an angle \(\theta\) is given by:\[R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}\]For \(\theta = 210^\circ\), we convert the angle to radians because the trigonometric functions in most mathematical settings use radians:\[\theta = 210^\circ = \frac{210 \pi}{180} = \frac{7\pi}{6} \text{ radians}\]Substituting \(\theta = \frac{7\pi}{6}\) into the rotation matrix:\[R\left(\frac{7\pi}{6}\right) = \begin{pmatrix} \cos\left(\frac{7\pi}{6}\right) & -\sin\left(\frac{7\pi}{6}\right) \\ \sin\left(\frac{7\pi}{6}\right) & \cos\left(\frac{7\pi}{6}\right) \end{pmatrix}\]Using trigonometric identities:\[\cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}, \quad \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}\]Thus, the rotation matrix becomes:\[R\left(\frac{7\pi}{6}\right) = \begin{pmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{pmatrix}\]**Application:**To find where

At first we find the rotation matrix for the transformation and then we find the new position of the…

Answered: 1.) Find the standard matrix for…

1.) Find the standard matrix for counterclockwise rotation of 210° about the origin in R2. Then use this to find where the point (5,8) goes under this transformation.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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I'm not sure how to do the counter clockwise rotations from the origin in this example?

**Topic: Linear Transformations in Euclidean Space**

### Problem Statement:

1.) Find the standard matrix for a counterclockwise rotation of \(210^\circ\) about the origin in \(\mathbb{R}^2\). Then use this to determine where the point \((5, 8)\) is mapped under this transformation.

**Explanation:**

To solve this problem, we need to construct the rotation matrix for a counterclockwise rotation by an angle \(\theta\) in the 2D plane. The standard rotation matrix for an angle \(\theta\) is given by:

\[
R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}
\]

For \(\theta = 210^\circ\), we convert the angle to radians because the trigonometric functions in most mathematical settings use radians:

\[
\theta = 210^\circ = \frac{210 \pi}{180} = \frac{7\pi}{6} \text{ radians}
\]

Substituting \(\theta = \frac{7\pi}{6}\) into the rotation matrix:

\[
R\left(\frac{7\pi}{6}\right) = \begin{pmatrix} \cos\left(\frac{7\pi}{6}\right) & -\sin\left(\frac{7\pi}{6}\right) \\ \sin\left(\frac{7\pi}{6}\right) & \cos\left(\frac{7\pi}{6}\right) \end{pmatrix}
\]

Using trigonometric identities:

\[
\cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}, \quad \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}
\]

Thus, the rotation matrix becomes:

\[
R\left(\frac{7\pi}{6}\right) = \begin{pmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{pmatrix}
\]

**Application:**

To find where

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