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Write v as the sum of two vector components if v = 2i + j and w = 3i − 2j. Ov(음+음)+(2+) O(+) O ..(금) (+) V=

Write v as the sum of two vector components if v = 2i + j and w = 3i − 2j. Ov(음+음)+(2+) O(+) O ..(금) (+) V=

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Vector Decomposition Problem **Problem Statement:** Write **v** as the sum of two vector components if **v** = 2**i** + **j** and **w** = 3**i** - 2**j**. **Answer Options:** 1. \( \mathbf{v} = \left( \frac{12}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) + \left( \frac{14}{13} \mathbf{i} + \frac{21}{13} \mathbf{j} \right) \) 2. \( \mathbf{v} = \left( \frac{12}{13} \mathbf{i} - \frac{8}{13} \mathbf{j} \right) + \left( \frac{14}{13} \mathbf{i} + \frac{21}{13} \mathbf{j} \right) \) 3. \( \mathbf{v} = \left( - \frac{1}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) + \left( \frac{14}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) \) 4. \( \mathbf{v} = \left( - \frac{1}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) + \left( \frac{2}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) \) **Explanation:** To solve this problem, we need to express vector **v** in terms of two components. Given vectors **v** and **w**, we may need to use vector algebra to decompose **v** into two components that potentially involve the direction given by **w** and another orthogonal direction. The options provided represent possible ways to express **v** as the sum of two vector components. Each equation shows **v** being broken down into two parts, which when added together should yield the original vector **v**. Further analysis and computations can be carried out step by step to identify the correct representation of **v**.
Transcribed Image Text:### Vector Decomposition Problem **Problem Statement:** Write **v** as the sum of two vector components if **v** = 2**i** + **j** and **w** = 3**i** - 2**j**. **Answer Options:** 1. \( \mathbf{v} = \left( \frac{12}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) + \left( \frac{14}{13} \mathbf{i} + \frac{21}{13} \mathbf{j} \right) \) 2. \( \mathbf{v} = \left( \frac{12}{13} \mathbf{i} - \frac{8}{13} \mathbf{j} \right) + \left( \frac{14}{13} \mathbf{i} + \frac{21}{13} \mathbf{j} \right) \) 3. \( \mathbf{v} = \left( - \frac{1}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) + \left( \frac{14}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) \) 4. \( \mathbf{v} = \left( - \frac{1}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) + \left( \frac{2}{13} \mathbf{i} + \frac{8}{13} \mathbf{j} \right) \) **Explanation:** To solve this problem, we need to express vector **v** in terms of two components. Given vectors **v** and **w**, we may need to use vector algebra to decompose **v** into two components that potentially involve the direction given by **w** and another orthogonal direction. The options provided represent possible ways to express **v** as the sum of two vector components. Each equation shows **v** being broken down into two parts, which when added together should yield the original vector **v**. Further analysis and computations can be carried out step by step to identify the correct representation of **v**.
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