Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 9E
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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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