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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![### Physics Problems - Vector Magnitudes and Relative Velocity
#### Problem 3:
In the above problem, what is \(|\vec{D}|\), the magnitude of vector \(\vec{D} = \vec{A} - \vec{B}\)?
- (A) 1.0 cm
- (B) 1.1 cm
- (C) 1.2 cm
- (D) 1.4 cm
#### Problem 4:
A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. What is the velocity relative to an observer on the shore?
- (A) 3.2 m/s to the southeast
- (B) 5.0 m/s to the southeast
- (C) 7.1 m/s to the southeast
- (D) 10.0 m/s to the southeast
### Explanation:
**For Problem 3:**
The problem requires determining the magnitude of the resultant vector \(\vec{D}\) when vector \(\vec{B}\) is subtracted from vector \(\vec{A}\). The correct answer can be chosen from options (A) to (D) based on the given numerical values.
**For Problem 4:**
The problem states that a motorboat moves due east at 5.0 m/s while a river flows south at 5.0 m/s. To find the velocity of the boat relative to an observer on the shore, we need to consider both velocities as perpendicular components of the resultant velocity.
- The eastward velocity component: \(\vec{v}_\text{boat} = 5.0 \, \text{m/s}\)
- The southward velocity component: \(\vec{v}_\text{river} = 5.0 \, \text{m/s}\)
Using the Pythagorean theorem:
\[ v_{\text{resultant}} = \sqrt{(v_{\text{boat}})^2 + (v_{\text{river}})^2} \]
\[ v_{\text{resultant}} = \sqrt{(5.0 \, \text{m/s})^2 + (5.0 \, \text{m/s})^2} \]
\[ v_{\text{resultant}} = \sqrt{25 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F838861be-5389-4021-b2a9-c2247c81cbf2%2F0e06e412-0d1f-453c-94d0-22e395d71348%2Fxcl022_processed.png&w=3840&q=75)
Transcribed Image Text:### Physics Problems - Vector Magnitudes and Relative Velocity
#### Problem 3:
In the above problem, what is \(|\vec{D}|\), the magnitude of vector \(\vec{D} = \vec{A} - \vec{B}\)?
- (A) 1.0 cm
- (B) 1.1 cm
- (C) 1.2 cm
- (D) 1.4 cm
#### Problem 4:
A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. What is the velocity relative to an observer on the shore?
- (A) 3.2 m/s to the southeast
- (B) 5.0 m/s to the southeast
- (C) 7.1 m/s to the southeast
- (D) 10.0 m/s to the southeast
### Explanation:
**For Problem 3:**
The problem requires determining the magnitude of the resultant vector \(\vec{D}\) when vector \(\vec{B}\) is subtracted from vector \(\vec{A}\). The correct answer can be chosen from options (A) to (D) based on the given numerical values.
**For Problem 4:**
The problem states that a motorboat moves due east at 5.0 m/s while a river flows south at 5.0 m/s. To find the velocity of the boat relative to an observer on the shore, we need to consider both velocities as perpendicular components of the resultant velocity.
- The eastward velocity component: \(\vec{v}_\text{boat} = 5.0 \, \text{m/s}\)
- The southward velocity component: \(\vec{v}_\text{river} = 5.0 \, \text{m/s}\)
Using the Pythagorean theorem:
\[ v_{\text{resultant}} = \sqrt{(v_{\text{boat}})^2 + (v_{\text{river}})^2} \]
\[ v_{\text{resultant}} = \sqrt{(5.0 \, \text{m/s})^2 + (5.0 \, \text{m/s})^2} \]
\[ v_{\text{resultant}} = \sqrt{25 +

Transcribed Image Text:The image displays a coordinate system with the \(x\)-axis and \(y\)-axis.
Two vectors are represented in this system:
1. Vector \(\vec{A}\) is positioned along the positive \(x\)-axis and has a magnitude of 2.0 cm.
2. Vector \(\vec{B}\) is oriented at an angle \(\alpha = 30^\circ\) relative to the positive \(x\)-axis. This vector also has a magnitude of 2.0 cm.
The image shows that the tail of each vector is positioned at the origin (0,0) of the coordinate system. Vector \(\vec{A}\) extends horizontally to the right, while vector \(\vec{B}\) extends diagonally upward to the right, forming a 30-degree angle with vector \(\vec{A}\).
This diagram can be useful for understanding the representation of vectors in a coordinate system, the concept of vector magnitude and direction, and how trigonometric functions can be applied to decompose vectors into their components.
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