College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![### Educational Content: Vectors and Their Magnitudes
**Problem Statement:**
If \( \mathbf{C} = [10 \, \text{m}, 30^\circ] \) and \( \mathbf{D} = [25 \, \text{m}, 130^\circ] \), what is the magnitude of the sum of these two vectors? (\(\mathbf{C}\) and \(\mathbf{D}\) are vectors)
**Options:**
- \( \, \, \circ \) 20 m
- \( \, \, \circ \) 35 m
- \( \, \, \circ \) 25 m
- \( \, \, \circ \) 15 m
**Explanation:**
To find the magnitude of the sum of two vectors \(\mathbf{C}\) and \(\mathbf{D}\), we first need to decompose each vector into its horizontal and vertical components. Then, we add the corresponding components together and calculate the resultant vector's magnitude.
1. **Decompose the vectors:**
- For vector \(\mathbf{C}\):
\[
\text{Horizontal Component} = 10 \, \text{m} \cdot \cos(30^\circ)
\]
\[
\text{Vertical Component} = 10 \, \text{m} \cdot \sin(30^\circ)
\]
- For vector \(\mathbf{D}\):
\[
\text{Horizontal Component} = 25 \, \text{m} \cdot \cos(130^\circ)
\]
\[
\text{Vertical Component} = 25 \, \text{m} \cdot \sin(130^\circ)
\]
2. **Add the components:**
- Sum of horizontal components:
\[
\text{Horizontal Sum} = (10 \, \text{m} \cdot \cos(30^\circ)) + (25 \, \text{m} \cdot \cos(130^\circ))
\]
- Sum of vertical components:
\[
\text{Vertical Sum} = (10 \, \text{m} \cdot \sin(30^\circ)) + (25 \, \text{m} \cdot \sin(130^\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb36efbe6-1aa0-4424-888d-f67fe8307a17%2F9d1ea127-064c-4bbe-a583-c71e8ede5e91%2Fgpokz2h_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content: Vectors and Their Magnitudes
**Problem Statement:**
If \( \mathbf{C} = [10 \, \text{m}, 30^\circ] \) and \( \mathbf{D} = [25 \, \text{m}, 130^\circ] \), what is the magnitude of the sum of these two vectors? (\(\mathbf{C}\) and \(\mathbf{D}\) are vectors)
**Options:**
- \( \, \, \circ \) 20 m
- \( \, \, \circ \) 35 m
- \( \, \, \circ \) 25 m
- \( \, \, \circ \) 15 m
**Explanation:**
To find the magnitude of the sum of two vectors \(\mathbf{C}\) and \(\mathbf{D}\), we first need to decompose each vector into its horizontal and vertical components. Then, we add the corresponding components together and calculate the resultant vector's magnitude.
1. **Decompose the vectors:**
- For vector \(\mathbf{C}\):
\[
\text{Horizontal Component} = 10 \, \text{m} \cdot \cos(30^\circ)
\]
\[
\text{Vertical Component} = 10 \, \text{m} \cdot \sin(30^\circ)
\]
- For vector \(\mathbf{D}\):
\[
\text{Horizontal Component} = 25 \, \text{m} \cdot \cos(130^\circ)
\]
\[
\text{Vertical Component} = 25 \, \text{m} \cdot \sin(130^\circ)
\]
2. **Add the components:**
- Sum of horizontal components:
\[
\text{Horizontal Sum} = (10 \, \text{m} \cdot \cos(30^\circ)) + (25 \, \text{m} \cdot \cos(130^\circ))
\]
- Sum of vertical components:
\[
\text{Vertical Sum} = (10 \, \text{m} \cdot \sin(30^\circ)) + (25 \, \text{m} \cdot \sin(130^\
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