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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![### Problem Statement
**Exercise 4:** Determine the angle between the vectors **v** and **w**.
Given vectors:
- \( v = \langle 1, 2, 3, 4 \rangle \)
- \( w = \langle 0, -1, 4, -2 \rangle \)
---
To find the angle \( \theta \) between the two vectors, you can use the dot product formula:
\[ \vec{v} \cdot \vec{w} = \|\vec{v}\| \|\vec{w}\| \cos(\theta) \]
Where:
- \( \vec{v} \cdot \vec{w} \) is the dot product of vectors \(v\) and \(w\).
- \( \|\vec{v}\| \) and \( \|\vec{w}\| \) are the magnitudes (or lengths) of vectors \(v\) and \(w\) respectively.
- \( \theta \) is the angle between the two vectors.
To solve this:
1. Find the dot product of \(v\) and \(w\).
2. Calculate the magnitudes of \(v\) and \(w\).
3. Use the dot product and magnitudes to find the cosine of the angle.
4. Solve for \( \theta \) using the arccos function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e6f2fcb-4128-4479-819e-77e934e4be0f%2F39c10f59-1e02-4d7f-8be5-93051a8b723d%2Fe5glkxl.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Exercise 4:** Determine the angle between the vectors **v** and **w**.
Given vectors:
- \( v = \langle 1, 2, 3, 4 \rangle \)
- \( w = \langle 0, -1, 4, -2 \rangle \)
---
To find the angle \( \theta \) between the two vectors, you can use the dot product formula:
\[ \vec{v} \cdot \vec{w} = \|\vec{v}\| \|\vec{w}\| \cos(\theta) \]
Where:
- \( \vec{v} \cdot \vec{w} \) is the dot product of vectors \(v\) and \(w\).
- \( \|\vec{v}\| \) and \( \|\vec{w}\| \) are the magnitudes (or lengths) of vectors \(v\) and \(w\) respectively.
- \( \theta \) is the angle between the two vectors.
To solve this:
1. Find the dot product of \(v\) and \(w\).
2. Calculate the magnitudes of \(v\) and \(w\).
3. Use the dot product and magnitudes to find the cosine of the angle.
4. Solve for \( \theta \) using the arccos function.
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