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Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Vector Field Formulation with Specific Properties
**Problem Statement:**
Write formulas for the vector field with the given properties. (Assume the vector field is two-dimensional.)
**Given:**
- All vectors point toward the origin and have a constant length.
### Explanation:
In this problem, we are asked to find the equations of a vector field where each vector points toward the origin and has a constant magnitude.
Let's denote the vector field \( \mathbf{F}(x,y) \).
**Step-by-Step Solution:**
1. **Direction Toward the Origin:**
- The direction vector of a point \( (x, y) \) toward the origin, \( (0, 0) \), is \( -\mathbf{r} = -\langle x, y \rangle \).
2. **Unit Vector Calculation:**
- Convert the direction vector into a unit vector:
\[
\mathbf{\hat{r}} = \frac{-\mathbf{r}}{|\mathbf{r}|} = \frac{-\langle x, y \rangle}{\sqrt{x^2 + y^2}}
\]
3. **Constant Vector Magnitude:**
- Let the constant magnitude be \( c \). The vector field should have a magnitude \( c \) in the direction toward the origin:
\[
\mathbf{F}(x, y) = c \mathbf{\hat{r}} = c \left(-\frac{\langle x, y \rangle}{\sqrt{x^2 + y^2}}\right) = -c\frac{\langle x, y \rangle}{\sqrt{x^2 + y^2}}
\]
4. **Resulting Vector Field:**
- Therefore, the vector field equations are:
\[
\mathbf{F}(x, y) = \left( -c\frac{x}{\sqrt{x^2 + y^2}}, -c\frac{y}{\sqrt{x^2 + y^2}} \right)
\]
This component form ensures that all vectors point toward the origin and have a constant length \( c \).
---
This text is designed to help students understand the process of determining a vector field with specific properties by breaking down the given problem into manageable steps and providing detailed mathematical derivations.](https://content.bartleby.com/qna-images/question/4d78e1cf-029c-45e6-87b1-24897302038c/7687ff92-e017-461d-8dd6-6970153e720a/tu2ii9n_thumbnail.jpeg)
Transcribed Image Text:### Vector Field Formulation with Specific Properties
**Problem Statement:**
Write formulas for the vector field with the given properties. (Assume the vector field is two-dimensional.)
**Given:**
- All vectors point toward the origin and have a constant length.
### Explanation:
In this problem, we are asked to find the equations of a vector field where each vector points toward the origin and has a constant magnitude.
Let's denote the vector field \( \mathbf{F}(x,y) \).
**Step-by-Step Solution:**
1. **Direction Toward the Origin:**
- The direction vector of a point \( (x, y) \) toward the origin, \( (0, 0) \), is \( -\mathbf{r} = -\langle x, y \rangle \).
2. **Unit Vector Calculation:**
- Convert the direction vector into a unit vector:
\[
\mathbf{\hat{r}} = \frac{-\mathbf{r}}{|\mathbf{r}|} = \frac{-\langle x, y \rangle}{\sqrt{x^2 + y^2}}
\]
3. **Constant Vector Magnitude:**
- Let the constant magnitude be \( c \). The vector field should have a magnitude \( c \) in the direction toward the origin:
\[
\mathbf{F}(x, y) = c \mathbf{\hat{r}} = c \left(-\frac{\langle x, y \rangle}{\sqrt{x^2 + y^2}}\right) = -c\frac{\langle x, y \rangle}{\sqrt{x^2 + y^2}}
\]
4. **Resulting Vector Field:**
- Therefore, the vector field equations are:
\[
\mathbf{F}(x, y) = \left( -c\frac{x}{\sqrt{x^2 + y^2}}, -c\frac{y}{\sqrt{x^2 + y^2}} \right)
\]
This component form ensures that all vectors point toward the origin and have a constant length \( c \).
---
This text is designed to help students understand the process of determining a vector field with specific properties by breaking down the given problem into manageable steps and providing detailed mathematical derivations.
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