6) cet Let w and ✓ be two vectors in kn with the property. that Proju w = = = and Projur=-w. what is the angle (in (in degrees) between wand V?
6) cet Let w and ✓ be two vectors in kn with the property. that Proju w = = = and Projur=-w. what is the angle (in (in degrees) between wand V?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 6:**
Given the vectors \( u \) and \( v \) in \( \mathbb{R}^n \) with the property that \(\text{proj}_v u = \frac{1}{4}v\) and \(\text{proj}_u v = -u\).
What is the angle (in degrees) between \( u \) and \( v \)?
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Explanation:
- \( \text{proj}_v u \) is the projection of vector \( u \) onto vector \( v \).
- Similarly, \( \text{proj}_u v \) is the projection of vector \( v \) onto vector \( u \).
This problem requires knowledge of vector projections and the relationship between the angle of two vectors and their dot product. Specifically, you may need to use the fact that the dot product of two vectors can be represented as \( u \cdot v = \|u\| \|v\| \cos(\theta) \), where \( \theta \) is the angle between \( u \) and \( v \).
For the projection formula:
\[ \text{proj}_v u = \frac{u \cdot v}{v \cdot v} v = \frac{1}{4}v \]
\[ \text{proj}_u v = \frac{u \cdot v}{u \cdot u} u = -u \]
Using these properties, students need to determine the angle \( \theta \) between the vectors \( u \) and \( v \).
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Transcribed Image Text:---
**Problem 6:**
Given the vectors \( u \) and \( v \) in \( \mathbb{R}^n \) with the property that \(\text{proj}_v u = \frac{1}{4}v\) and \(\text{proj}_u v = -u\).
What is the angle (in degrees) between \( u \) and \( v \)?
---
Explanation:
- \( \text{proj}_v u \) is the projection of vector \( u \) onto vector \( v \).
- Similarly, \( \text{proj}_u v \) is the projection of vector \( v \) onto vector \( u \).
This problem requires knowledge of vector projections and the relationship between the angle of two vectors and their dot product. Specifically, you may need to use the fact that the dot product of two vectors can be represented as \( u \cdot v = \|u\| \|v\| \cos(\theta) \), where \( \theta \) is the angle between \( u \) and \( v \).
For the projection formula:
\[ \text{proj}_v u = \frac{u \cdot v}{v \cdot v} v = \frac{1}{4}v \]
\[ \text{proj}_u v = \frac{u \cdot v}{u \cdot u} u = -u \]
Using these properties, students need to determine the angle \( \theta \) between the vectors \( u \) and \( v \).
---
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