Write formulas for the vector field with the given properties. (Assume the vector field is two dimensional.) All vectors point toward the origin and have constant length.
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
![### 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 \( c \).
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d78e1cf-029c-45e6-87b1-24897302038c%2F7687ff92-e017-461d-8dd6-6970153e720a%2Ftu2ii9n_processed.jpeg&w=3840&q=75)

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