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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Question
![**Question:**
Given \(\mathbf{v} = (2, -9)\) and \(\mathbf{w} = (7, -8)\), what is \(2\mathbf{w} - \mathbf{v}\) written as a linear combination of unit vectors?
**Explanation:**
To solve this problem, we will first compute the vector \(2\mathbf{w}\) and then subtract vector \(\mathbf{v}\) from it. After that, we will express the resultant vector as a linear combination of unit vectors \(\hat{i}\) and \(\hat{j}\), where \(\hat{i}\) represents the unit vector in the x-direction and \(\hat{j}\) represents the unit vector in the y-direction.
**Step-by-Step Solution:**
1. **Find \(2\mathbf{w}\):**
\[
2\mathbf{w} = 2 \cdot (7, -8) = (2 \cdot 7, 2 \cdot -8) = (14, -16)
\]
2. **Subtract \(\mathbf{v}\) from \(2\mathbf{w}\):**
\[
2\mathbf{w} - \mathbf{v} = (14, -16) - (2, -9) = (14 - 2, -16 - (-9)) = (12, -7)
\]
3. **Express the resultant vector as a linear combination of unit vectors:**
\[
2\mathbf{w} - \mathbf{v} = 12\hat{i} - 7\hat{j}
\]
So, the vector \(2\mathbf{w} - \mathbf{v}\) as a linear combination of unit vectors is:
\[
12\hat{i} - 7\hat{j}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09573b81-484b-4c4c-bf86-f91c298bbebe%2F70d3e2cb-fa76-475a-baf0-f107abc4030a%2F50kzw6v_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Given \(\mathbf{v} = (2, -9)\) and \(\mathbf{w} = (7, -8)\), what is \(2\mathbf{w} - \mathbf{v}\) written as a linear combination of unit vectors?
**Explanation:**
To solve this problem, we will first compute the vector \(2\mathbf{w}\) and then subtract vector \(\mathbf{v}\) from it. After that, we will express the resultant vector as a linear combination of unit vectors \(\hat{i}\) and \(\hat{j}\), where \(\hat{i}\) represents the unit vector in the x-direction and \(\hat{j}\) represents the unit vector in the y-direction.
**Step-by-Step Solution:**
1. **Find \(2\mathbf{w}\):**
\[
2\mathbf{w} = 2 \cdot (7, -8) = (2 \cdot 7, 2 \cdot -8) = (14, -16)
\]
2. **Subtract \(\mathbf{v}\) from \(2\mathbf{w}\):**
\[
2\mathbf{w} - \mathbf{v} = (14, -16) - (2, -9) = (14 - 2, -16 - (-9)) = (12, -7)
\]
3. **Express the resultant vector as a linear combination of unit vectors:**
\[
2\mathbf{w} - \mathbf{v} = 12\hat{i} - 7\hat{j}
\]
So, the vector \(2\mathbf{w} - \mathbf{v}\) as a linear combination of unit vectors is:
\[
12\hat{i} - 7\hat{j}
\]
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