Consider the following. T is the clockwise rotation (0 is negative) of 60° in R2, v = (1, 6). (a) Find the standard matrix A for the linear transformation T. A = (b) Use A to find the image of the vector v. T(v) =
Consider the following. T is the clockwise rotation (0 is negative) of 60° in R2, v = (1, 6). (a) Find the standard matrix A for the linear transformation T. A = (b) Use A to find the image of the vector v. T(v) =
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter6: Matrices And Determinants
Section: Chapter Questions
Problem 5P
Related questions
Question
![### Linear Algebra Problem: Clockwise Rotation
Consider the following:
**T** is the clockwise rotation (θ is negative) of 60° in ℝ², **v** = (1, 6).
#### (a) Find the standard matrix **A** for the linear transformation **T**.
\[ A = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix} \]
#### (b) Use **A** to find the image of the vector **v**.
\[ T(v) = \boxed{} \]
---
In this problem, we are dealing with a 2-dimensional vector transformation involving a clockwise rotation by 60°.
1. **Standard Matrix for Rotation**:
- When θ = -60° (clockwise), the rotation matrix **A** in ℝ² is given by:
\[
A = \begin{bmatrix}
\cos(-60°) & -\sin(-60°) \\
\sin(-60°) & \cos(-60°)
\end{bmatrix}
\]
- Simplifying using trigonometric values:
\[
\cos(-60°) = \frac{1}{2}, \quad \sin(-60°) = -\frac{\sqrt{3}}{2}
\]
- Thus,
\[
A = \begin{bmatrix}
\frac{1}{2} & \frac{\sqrt{3}}{2} \\
-\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{bmatrix}
\]
2. **Finding the image of the vector**:
- The given vector **v** = (1, 6).
- To find the image **T(v)** under the transformation **T**, perform the matrix multiplication:
\[
T(v) = A \cdot v = \begin{bmatrix}
\frac{1}{2} & \frac{\sqrt{3}}{2} \\
-\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{bmatrix}
\begin{bmatrix}
1 \\
6
\end{bmatrix}
\]
- Calculate:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9d41188-65f6-48b0-af2f-be750049d164%2F4b455d3d-4d55-413f-8acb-6f9660ce41d3%2F2t4lyhn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem: Clockwise Rotation
Consider the following:
**T** is the clockwise rotation (θ is negative) of 60° in ℝ², **v** = (1, 6).
#### (a) Find the standard matrix **A** for the linear transformation **T**.
\[ A = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix} \]
#### (b) Use **A** to find the image of the vector **v**.
\[ T(v) = \boxed{} \]
---
In this problem, we are dealing with a 2-dimensional vector transformation involving a clockwise rotation by 60°.
1. **Standard Matrix for Rotation**:
- When θ = -60° (clockwise), the rotation matrix **A** in ℝ² is given by:
\[
A = \begin{bmatrix}
\cos(-60°) & -\sin(-60°) \\
\sin(-60°) & \cos(-60°)
\end{bmatrix}
\]
- Simplifying using trigonometric values:
\[
\cos(-60°) = \frac{1}{2}, \quad \sin(-60°) = -\frac{\sqrt{3}}{2}
\]
- Thus,
\[
A = \begin{bmatrix}
\frac{1}{2} & \frac{\sqrt{3}}{2} \\
-\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{bmatrix}
\]
2. **Finding the image of the vector**:
- The given vector **v** = (1, 6).
- To find the image **T(v)** under the transformation **T**, perform the matrix multiplication:
\[
T(v) = A \cdot v = \begin{bmatrix}
\frac{1}{2} & \frac{\sqrt{3}}{2} \\
-\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{bmatrix}
\begin{bmatrix}
1 \\
6
\end{bmatrix}
\]
- Calculate:
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
