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,...
Related questions
Question
![**Title: Calculating the Angle Between Two Vectors**
**Objective:**
Learn how to find the angle \( \theta \) between two vectors.
**Problem Statement:**
Find the angle \( \theta \) between the vectors \( \mathbf{u} \) and \( \mathbf{v} \). Round your answer to two decimal places.
**Vectors Provided:**
\[ \mathbf{u} = \begin{pmatrix} -3 \\ -5 \end{pmatrix} \]
\[ \mathbf{v} = \begin{pmatrix} 0 \\ -4 \end{pmatrix} \]
**Formula:**
\[ (\mathbf{u}, \mathbf{v}) = u_1v_1 + u_2v_2 \]
\[ \theta = \boxed{\phantom{xx}} \text{ radians} \]
### Step-by-Step Solution
1. **Dot Product:**
Calculate the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \).
\[ \mathbf{u} \cdot \mathbf{v} = (-3)(0) + (-5)(-4) \]
2. **Magnitude of Vectors:**
Find the magnitudes of each vector.
\[ |\mathbf{u}| = \sqrt{(-3)^2 + (-5)^2} \]
\[ |\mathbf{v}| = \sqrt{(0)^2 + (-4)^2} \]
3. **Cosine of Angle:**
Use the dot product and magnitudes to find cosine of the angle.
\[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \]
4. **Angle Calculation:**
Use the arccosine to find the angle \( \theta \).
\[ \theta = \arccos\left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \right) \]
### Detailed Graph/Diagram Explanation
In this problem, there isn’t a specific graph or diagram provided. However, for educational purposes, here is a brief description of the process of visualizing these calculations:
- **Vector Representation:**
- The vectors \( \mathbf{u](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F031412a4-ce24-4200-b630-467aaa8d4c51%2Fe5a99f57-4362-4423-ba8a-f04780a493d3%2Fwo7z7px_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Angle Between Two Vectors**
**Objective:**
Learn how to find the angle \( \theta \) between two vectors.
**Problem Statement:**
Find the angle \( \theta \) between the vectors \( \mathbf{u} \) and \( \mathbf{v} \). Round your answer to two decimal places.
**Vectors Provided:**
\[ \mathbf{u} = \begin{pmatrix} -3 \\ -5 \end{pmatrix} \]
\[ \mathbf{v} = \begin{pmatrix} 0 \\ -4 \end{pmatrix} \]
**Formula:**
\[ (\mathbf{u}, \mathbf{v}) = u_1v_1 + u_2v_2 \]
\[ \theta = \boxed{\phantom{xx}} \text{ radians} \]
### Step-by-Step Solution
1. **Dot Product:**
Calculate the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \).
\[ \mathbf{u} \cdot \mathbf{v} = (-3)(0) + (-5)(-4) \]
2. **Magnitude of Vectors:**
Find the magnitudes of each vector.
\[ |\mathbf{u}| = \sqrt{(-3)^2 + (-5)^2} \]
\[ |\mathbf{v}| = \sqrt{(0)^2 + (-4)^2} \]
3. **Cosine of Angle:**
Use the dot product and magnitudes to find cosine of the angle.
\[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \]
4. **Angle Calculation:**
Use the arccosine to find the angle \( \theta \).
\[ \theta = \arccos\left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \right) \]
### Detailed Graph/Diagram Explanation
In this problem, there isn’t a specific graph or diagram provided. However, for educational purposes, here is a brief description of the process of visualizing these calculations:
- **Vector Representation:**
- The vectors \( \mathbf{u
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 3 steps with 3 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education