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
![**Vectors in the Plane**
### Find the component form and magnitude of the vector \( \mathbf{v} \).
#### Inputs:
- **component form:**
\(\mathbf{v} = \text{[text box]}\)
- **magnitude:**
\(||\mathbf{v}|| = \text{[text box]}\)
### Graph Explanation:
A Cartesian coordinate system is displayed with both \(x\)- and \(y\)-axes labeled. A vector \( \mathbf{v} \) is depicted with its arrowhead at the point \((7, 7)\) and its tail at the point \((7, -3)\). The vector is vertically oriented.
#### Characteristics of the Vector:
- **Initial Point (Tail):** \((7, -3)\)
- **Terminal Point (Head):** \((7, 7)\)
### Graph Interpretation:
- The vector starts at the coordinates \((7, -3)\) and ends at \((7, 7)\).
- The \(x\)-coordinate of the vector is constant at \(7\), while the \(y\)-coordinate changes from \(-3\) to \(7\).
**Note:** The component form and magnitude of the vector \( \mathbf{v} \) can be found using the coordinates of the initial and terminal points.
### Calculations:
1. **Component Form:**
To find the component form, subtract the coordinates of the initial point from the coordinates of the terminal point.
\[
\mathbf{v} = (7 - 7, 7 - (-3)) = (0, 10)
\]
2. **Magnitude:**
To find the magnitude of the vector, use the formula:
\[
||\mathbf{v}|| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For the given points,
\[
||\mathbf{v}|| = \sqrt{(7 - 7)^2 + (7 - (-3))^2} = \sqrt{0 + 10^2} = \sqrt{100} = 10
\]
### Summary:
- **Component Form:** \( \mathbf{v} = (0, 10) \)
- **Magnitude:** \( ||\mathbf{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61a624e8-9805-4882-bd7c-8b439f9ba0a9%2Fd013428f-3dab-4fd3-a65a-7ebb15040a21%2Fyop7dxh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vectors in the Plane**
### Find the component form and magnitude of the vector \( \mathbf{v} \).
#### Inputs:
- **component form:**
\(\mathbf{v} = \text{[text box]}\)
- **magnitude:**
\(||\mathbf{v}|| = \text{[text box]}\)
### Graph Explanation:
A Cartesian coordinate system is displayed with both \(x\)- and \(y\)-axes labeled. A vector \( \mathbf{v} \) is depicted with its arrowhead at the point \((7, 7)\) and its tail at the point \((7, -3)\). The vector is vertically oriented.
#### Characteristics of the Vector:
- **Initial Point (Tail):** \((7, -3)\)
- **Terminal Point (Head):** \((7, 7)\)
### Graph Interpretation:
- The vector starts at the coordinates \((7, -3)\) and ends at \((7, 7)\).
- The \(x\)-coordinate of the vector is constant at \(7\), while the \(y\)-coordinate changes from \(-3\) to \(7\).
**Note:** The component form and magnitude of the vector \( \mathbf{v} \) can be found using the coordinates of the initial and terminal points.
### Calculations:
1. **Component Form:**
To find the component form, subtract the coordinates of the initial point from the coordinates of the terminal point.
\[
\mathbf{v} = (7 - 7, 7 - (-3)) = (0, 10)
\]
2. **Magnitude:**
To find the magnitude of the vector, use the formula:
\[
||\mathbf{v}|| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For the given points,
\[
||\mathbf{v}|| = \sqrt{(7 - 7)^2 + (7 - (-3))^2} = \sqrt{0 + 10^2} = \sqrt{100} = 10
\]
### Summary:
- **Component Form:** \( \mathbf{v} = (0, 10) \)
- **Magnitude:** \( ||\mathbf{
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