Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 11E
Related questions
Question
![### Problem Statement
Let \( w \) and \( v \) be two vectors in \( \mathbb{R}^n \) with the property that \( \text{Proj}_vw = -\frac{1}{4}v \) and \( \text{Proj}_wv = -w \). What is the angle (in degrees) between \( w \) and \( v \)?
\[ \boxed{} \]
### Explanation
- **Vectors in \( \mathbb{R}^n \)**: The vectors \( w \) and \( v \) are elements of n-dimensional real space.
- **Projection**: The notation \( \text{Proj}_vw \) denotes the projection of the vector \( w \) onto the vector \( v \). The given conditions are:
\[
\text{Proj}_vw = -\frac{1}{4}v
\]
and
\[
\text{Proj}_wv = -w.
\]
To solve for the angle between \( w \) and \( v \), one needs to use the definition of projection and the dot product properties.
- **Projection Formula**: If \( u \) and \( v \) are vectors in \( \mathbb{R}^n \), then the projection of \( u \) onto \( v \) is given by:
\[
\text{Proj}_vu = \frac{u \cdot v}{v \cdot v}v
\]
Where \( u \cdot v \) is the dot product of \( u \) and \( v \), and \( v \cdot v \) is the dot product of \( v \) with itself (which is equal to \( ||v||^2 \), the squared norm of \( v \)).
Given the conditions, you can use these properties to derive the necessary equations and solve for the angle between the vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe354d500-dc10-4e3d-a060-42009e500b8b%2Fd4c361a9-1210-4091-b943-9c7c101ab3bb%2Fdnemx8e_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Let \( w \) and \( v \) be two vectors in \( \mathbb{R}^n \) with the property that \( \text{Proj}_vw = -\frac{1}{4}v \) and \( \text{Proj}_wv = -w \). What is the angle (in degrees) between \( w \) and \( v \)?
\[ \boxed{} \]
### Explanation
- **Vectors in \( \mathbb{R}^n \)**: The vectors \( w \) and \( v \) are elements of n-dimensional real space.
- **Projection**: The notation \( \text{Proj}_vw \) denotes the projection of the vector \( w \) onto the vector \( v \). The given conditions are:
\[
\text{Proj}_vw = -\frac{1}{4}v
\]
and
\[
\text{Proj}_wv = -w.
\]
To solve for the angle between \( w \) and \( v \), one needs to use the definition of projection and the dot product properties.
- **Projection Formula**: If \( u \) and \( v \) are vectors in \( \mathbb{R}^n \), then the projection of \( u \) onto \( v \) is given by:
\[
\text{Proj}_vu = \frac{u \cdot v}{v \cdot v}v
\]
Where \( u \cdot v \) is the dot product of \( u \) and \( v \), and \( v \cdot v \) is the dot product of \( v \) with itself (which is equal to \( ||v||^2 \), the squared norm of \( v \)).
Given the conditions, you can use these properties to derive the necessary equations and solve for the angle between the vectors.
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