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Use the vectors shown to sketch u - 3v. (Assume that each point lies on the grid lines and that each grid line is spaced one unit apart.) и

Use the vectors shown to sketch u - 3v. (Assume that each point lies on the grid lines and that each grid line is spaced one unit apart.) и

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
**Educational Content: Vectors and Linear Combinations** **Objective:** Learn how to sketch the vector \( \mathbf{u} - 3\mathbf{v} \) using given vectors \( \mathbf{u} \) and \( \mathbf{v} \). **Instructions:** Use the vectors shown to sketch \( \mathbf{u} - 3\mathbf{v} \). Assume that each point lies on the grid lines and each grid line is spaced one unit apart. **Graphs and Diagrams Explanation:** 1. **Initial Diagram:** - Vectors \( \mathbf{u} \) and \( \mathbf{v} \) are displayed on a grid, each starting from a common point. - Vector \( \mathbf{u} \) is slightly longer and angled upward to the right. - Vector \( \mathbf{v} \) is shorter and angled downward to the left. 2. **Four Graph Options:** - These graphs represent possible visualizations of the resultant vector \( \mathbf{u} - 3\mathbf{v} \). - **Option A:** Shows a medium-length vector angled slightly upward. - **Option B:** Shows a shorter vector angled downward to the left. - **Option C:** Illustrates a longer vector angled downward. - **Option D:** Depicts a vector with a steep upward angle and moderate length. **Key Concept:** To determine \( \mathbf{u} - 3\mathbf{v} \), you need to: 1. Multiply vector \( \mathbf{v} \) by 3 (extend its length three times in the same direction). 2. Subtract this resultant vector from \( \mathbf{u} \). Visualize this operation by placing the tail of \( 3\mathbf{v} \) at the head of \( \mathbf{u} \) and drawing the resultant vector from the tail of \( \mathbf{u} \) to the head of \( -3\mathbf{v} \). **Conclusion:** Understanding vector operations such as addition and subtraction in geometric terms helps visualize and solve vector problems more effectively. Use the correct option to depict the accurate vector \( \mathbf{u} - 3\mathbf{v} \).
Transcribed Image Text:**Educational Content: Vectors and Linear Combinations** **Objective:** Learn how to sketch the vector \( \mathbf{u} - 3\mathbf{v} \) using given vectors \( \mathbf{u} \) and \( \mathbf{v} \). **Instructions:** Use the vectors shown to sketch \( \mathbf{u} - 3\mathbf{v} \). Assume that each point lies on the grid lines and each grid line is spaced one unit apart. **Graphs and Diagrams Explanation:** 1. **Initial Diagram:** - Vectors \( \mathbf{u} \) and \( \mathbf{v} \) are displayed on a grid, each starting from a common point. - Vector \( \mathbf{u} \) is slightly longer and angled upward to the right. - Vector \( \mathbf{v} \) is shorter and angled downward to the left. 2. **Four Graph Options:** - These graphs represent possible visualizations of the resultant vector \( \mathbf{u} - 3\mathbf{v} \). - **Option A:** Shows a medium-length vector angled slightly upward. - **Option B:** Shows a shorter vector angled downward to the left. - **Option C:** Illustrates a longer vector angled downward. - **Option D:** Depicts a vector with a steep upward angle and moderate length. **Key Concept:** To determine \( \mathbf{u} - 3\mathbf{v} \), you need to: 1. Multiply vector \( \mathbf{v} \) by 3 (extend its length three times in the same direction). 2. Subtract this resultant vector from \( \mathbf{u} \). Visualize this operation by placing the tail of \( 3\mathbf{v} \) at the head of \( \mathbf{u} \) and drawing the resultant vector from the tail of \( \mathbf{u} \) to the head of \( -3\mathbf{v} \). **Conclusion:** Understanding vector operations such as addition and subtraction in geometric terms helps visualize and solve vector problems more effectively. Use the correct option to depict the accurate vector \( \mathbf{u} - 3\mathbf{v} \).
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