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
![### Calculating the Dot Product of Two Vectors
#### Problem Statement
Given vectors:
\[ \mathbf{v} = 9\mathbf{i} - 5\mathbf{j} \]
\[ \mathbf{w} = -3\mathbf{i} - 4\mathbf{j} \]
Find the dot product \(\mathbf{v} \cdot \mathbf{w}\).
#### Options
- \( 59 \)
- \( 7 \)
- \( -59 \)
- \( -7 \)
#### Solution
To calculate the dot product of two vectors \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{w} = c\mathbf{i} + d\mathbf{j}\), use the formula:
\[ \mathbf{v} \cdot \mathbf{w} = ac + bd \]
Given:
\[ \mathbf{v} = 9\mathbf{i} - 5\mathbf{j} \]
\[ \mathbf{w} = -3\mathbf{i} - 4\mathbf{j} \]
Plugging in the values:
\[ \mathbf{v} \cdot \mathbf{w} = (9)(-3) + (-5)(-4) \]
\[ \mathbf{v} \cdot \mathbf{w} = -27 + 20 \]
\[ \mathbf{v} \cdot \mathbf{w} = -7 \]
Therefore, the correct answer is:
- \( -7 \)
#### Answer
\( -7 \)
#### Diagram Explanation
There is no diagram or graph provided in the problem.
---
**Note:**
This type of problem helps in understanding vector operations which are fundamental in physics and engineering. The dot product, in particular, is widely used in projections, work calculations, and determining angles between vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc5bcc65-7283-4529-b9b7-ffbb90a5afc1%2F87df0582-d113-4e00-ad4e-f0750e0393a7%2Fyg4to2b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Dot Product of Two Vectors
#### Problem Statement
Given vectors:
\[ \mathbf{v} = 9\mathbf{i} - 5\mathbf{j} \]
\[ \mathbf{w} = -3\mathbf{i} - 4\mathbf{j} \]
Find the dot product \(\mathbf{v} \cdot \mathbf{w}\).
#### Options
- \( 59 \)
- \( 7 \)
- \( -59 \)
- \( -7 \)
#### Solution
To calculate the dot product of two vectors \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{w} = c\mathbf{i} + d\mathbf{j}\), use the formula:
\[ \mathbf{v} \cdot \mathbf{w} = ac + bd \]
Given:
\[ \mathbf{v} = 9\mathbf{i} - 5\mathbf{j} \]
\[ \mathbf{w} = -3\mathbf{i} - 4\mathbf{j} \]
Plugging in the values:
\[ \mathbf{v} \cdot \mathbf{w} = (9)(-3) + (-5)(-4) \]
\[ \mathbf{v} \cdot \mathbf{w} = -27 + 20 \]
\[ \mathbf{v} \cdot \mathbf{w} = -7 \]
Therefore, the correct answer is:
- \( -7 \)
#### Answer
\( -7 \)
#### Diagram Explanation
There is no diagram or graph provided in the problem.
---
**Note:**
This type of problem helps in understanding vector operations which are fundamental in physics and engineering. The dot product, in particular, is widely used in projections, work calculations, and determining angles between vectors.
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