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Your friend Adrian is 5 ft from you—call Adrian’s position vector from you " vector A"⃑. Your friend Becky is 3 ft from Adrian—call Becky’s position vector from Adrian " Vector B".

Your friend Adrian is 5 ft from you—call Adrian’s position vector from you " vector A"⃑. Your friend Becky is 3 ft from Adrian—call Becky’s position vector from Adrian " Vector B".

College Physics
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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Your friend Adrian is 5 ft from you—call Adrian’s position vector from you " vector A"⃑. Your friend Becky is 3 ft from Adrian—call Becky’s position vector from Adrian "
Vector B".

*First image is questioned, and the second image(my solution) is my professor's feedback. 1-c please write your solution in as detail as possible.

### Problem 1: #### Context: - **Adrian** is 5 ft from you, noted as position vector \( \vec{A} \). - **Becky** is 3 ft from Adrian, noted as position vector \( \vec{B} \). - Define positions relative to reference points \( P, Q, \) and \( R \). #### Questions: **a.** For the **farthest distance** Becky can be from you, based on vectors, with details on angle: - **Distance Calculations**: \[ |\vec{R}| = 8 \text{ ft} \] - **Vector Sketch**: \( \vec{A} \) starts from \( P \) to \( Q \), and \( \vec{B} \) from \( Q \) to \( R \). - **Angle**: \[ \theta = 0^\circ \] **b.** For the **shortest distance** Becky can be from you: - **Distance Calculations**: \[ |\vec{R}| = 2 \text{ ft} \] - **Vector Sketch**: \( \vec{A} \) from \( P \) to \( Q \), \( \vec{B} \) from \( Q \) to \( R \). - **Angle**: \[ \theta = 180^\circ \] **c.** If Becky and Adrian are at **equal distances** from you: - **Distance Calculations**: \[ |\vec{R}| = 5\sqrt{2} \text{ ft} \] - **Vector Sketch**: Showing positions \( P, R, Q \). - **Angle**: \[ \theta = \cos^{-1}\left(\frac{-3}{10}\right) \] #### Calculations: - Reviews and verifies using vector algebra without approximations. - Ensures validity of shortest and longest paths through calculations. ### Additional Notes: Corrections after problem-solving discussions were conducted and suggestions were provided: - **Concern Regarding Angles**: Highlighted consideration of analysis-related angles. - **Formula Errors**: Identified and recommended checking the used formulas. This analysis is intended to assist in understanding vector positioning and calculating distances and angles in two-dimensional space through theoretical scenarios.
Transcribed Image Text:### Problem 1: #### Context: - **Adrian** is 5 ft from you, noted as position vector \( \vec{A} \). - **Becky** is 3 ft from Adrian, noted as position vector \( \vec{B} \). - Define positions relative to reference points \( P, Q, \) and \( R \). #### Questions: **a.** For the **farthest distance** Becky can be from you, based on vectors, with details on angle: - **Distance Calculations**: \[ |\vec{R}| = 8 \text{ ft} \] - **Vector Sketch**: \( \vec{A} \) starts from \( P \) to \( Q \), and \( \vec{B} \) from \( Q \) to \( R \). - **Angle**: \[ \theta = 0^\circ \] **b.** For the **shortest distance** Becky can be from you: - **Distance Calculations**: \[ |\vec{R}| = 2 \text{ ft} \] - **Vector Sketch**: \( \vec{A} \) from \( P \) to \( Q \), \( \vec{B} \) from \( Q \) to \( R \). - **Angle**: \[ \theta = 180^\circ \] **c.** If Becky and Adrian are at **equal distances** from you: - **Distance Calculations**: \[ |\vec{R}| = 5\sqrt{2} \text{ ft} \] - **Vector Sketch**: Showing positions \( P, R, Q \). - **Angle**: \[ \theta = \cos^{-1}\left(\frac{-3}{10}\right) \] #### Calculations: - Reviews and verifies using vector algebra without approximations. - Ensures validity of shortest and longest paths through calculations. ### Additional Notes: Corrections after problem-solving discussions were conducted and suggestions were provided: - **Concern Regarding Angles**: Highlighted consideration of analysis-related angles. - **Formula Errors**: Identified and recommended checking the used formulas. This analysis is intended to assist in understanding vector positioning and calculating distances and angles in two-dimensional space through theoretical scenarios.
Here is a transcription and description suitable for an educational website: --- **Problem 1: Vector Analysis with Friends** You have two friends, Adrian and Becky. Consider the following scenarios based on their positions relative to you: - **Adrian's Position**: Adrian is 5 feet away from you. Represent this with the position vector \(\vec{A}\). - **Becky's Position**: Becky is 3 feet away from Adrian. Represent this with the position vector \(\vec{B}\) from Adrian to Becky. ### a. Farthest Distance Scenario Determine the farthest distance Becky could be from you. Sketch the vectors in this situation: - **Distance**: [Your answer here] - **Sketch**: [Insert sketch showing \(\vec{A}\) and \(\vec{B}\) vectors, with Becky positioned such that the vectors form a straight line, maximizing distance.] - **Angle**: [Your answer here, angle \(\vec{A}\) and \(\vec{B}\) in this configuration is 0 degrees.] ### b. Shortest Distance Scenario Determine the shortest distance Becky could be from you. Sketch the vectors in this situation: - **Distance**: [Your answer here] - **Sketch**: [Insert sketch showing \(\vec{A}\) and \(\vec{B}\) vectors with Becky positioned such that they form a straight line in the opposite direction, minimizing distance.] - **Angle**: [Your answer here, angle \(\vec{A}\) and \(\vec{B}\) in this configuration is 180 degrees.] ### c. Equal Distance Scenario If Becky is the same distance from you as Adrian is, sketch the vectors: - **Distance**: [Your calculations and answer here] - **Sketch**: [Insert sketch showing vectors \(\vec{A}\) and \(\vec{B}\) where Becky is at a 90-degree angle from Adrian, forming an isosceles triangle with the line from you to Becky.] - **Angle**: [Your answer here, angle is 90 degrees.] **Instructions:** Show detailed calculations for part (c), aiming for exact answers without approximations or calculators. --- This setup ensures clarity in understanding the geometric and vector relationships present in the problem.
Transcribed Image Text:Here is a transcription and description suitable for an educational website: --- **Problem 1: Vector Analysis with Friends** You have two friends, Adrian and Becky. Consider the following scenarios based on their positions relative to you: - **Adrian's Position**: Adrian is 5 feet away from you. Represent this with the position vector \(\vec{A}\). - **Becky's Position**: Becky is 3 feet away from Adrian. Represent this with the position vector \(\vec{B}\) from Adrian to Becky. ### a. Farthest Distance Scenario Determine the farthest distance Becky could be from you. Sketch the vectors in this situation: - **Distance**: [Your answer here] - **Sketch**: [Insert sketch showing \(\vec{A}\) and \(\vec{B}\) vectors, with Becky positioned such that the vectors form a straight line, maximizing distance.] - **Angle**: [Your answer here, angle \(\vec{A}\) and \(\vec{B}\) in this configuration is 0 degrees.] ### b. Shortest Distance Scenario Determine the shortest distance Becky could be from you. Sketch the vectors in this situation: - **Distance**: [Your answer here] - **Sketch**: [Insert sketch showing \(\vec{A}\) and \(\vec{B}\) vectors with Becky positioned such that they form a straight line in the opposite direction, minimizing distance.] - **Angle**: [Your answer here, angle \(\vec{A}\) and \(\vec{B}\) in this configuration is 180 degrees.] ### c. Equal Distance Scenario If Becky is the same distance from you as Adrian is, sketch the vectors: - **Distance**: [Your calculations and answer here] - **Sketch**: [Insert sketch showing vectors \(\vec{A}\) and \(\vec{B}\) where Becky is at a 90-degree angle from Adrian, forming an isosceles triangle with the line from you to Becky.] - **Angle**: [Your answer here, angle is 90 degrees.] **Instructions:** Show detailed calculations for part (c), aiming for exact answers without approximations or calculators. --- This setup ensures clarity in understanding the geometric and vector relationships present in the problem.
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