# Vector-Valued Function Ellipse

• vandersmissen
In summary, the given curve, 4x2+9y2=36, can be described by the vector-valued function 3cos(t)i+2sin(t)j in a counterclockwise direction, and by 3cos(t)i-2sin(t)j in a clockwise direction. This solution is derived by using the equation x2+y2=r2 and the trigonometric identity cos2t+sin2t=1.
vandersmissen

## Homework Statement

Find a vector-valued function f that traces out the given curve in the indicated direction.
(a) Counterclockwise (b) Clockwise.

4x2+9y2=36

x2+y2=r2
cos2t+sin2t=1

## The Attempt at a Solution

From what I can determine, this is an ellipse. I believe this is how the answer is found.

4x2/36+9y2/36=36/36

x2/9+y2/4=1

so cos2t+sin2t=1 and x2/9+y2/4=1
this means x=3cos(t) and y=2sin(t) to balance it equivalently, I think.

putting that in component form, 3cos(t)i+2sin(t)j for counterclockwise and 3cos(t)i-2sin(t)j for clockwise

That does equal the correct answer , but I am not sure if I got to it correctly. Can someone please review my work and let me know if I am approaching this correctly, I do not show anything in my notes or in my book on how to perform this kind of equation.

Yes, that is perfectly correct. Notice that from the definition of sine and cosine in terms of the unit circle, "t" goes around the circle counterclockwise so your 3cos(t)i+ 2sin(t)j does describe the circle counterclockwise as t increases. To go around the circle clockwise, use -t instead: 3cos(-t)+ 2sin(-t)= 3cos(t)- 2sin(-t) because cosine is an even function and sine is an odd function.

## 1. What is a vector-valued function ellipse?

A vector-valued function ellipse is a mathematical concept that describes a set of points in a two-dimensional space that forms an ellipse. It is defined by a vector-valued function, where the coordinates of each point on the ellipse are determined by a pair of input parameters.

## 2. How is a vector-valued function ellipse different from a regular ellipse?

A regular ellipse is defined by a set of two-dimensional coordinates, while a vector-valued function ellipse is defined by a vector-valued function. This means that a regular ellipse can be described using simple equations, while a vector-valued function ellipse requires more complex mathematical calculations.

## 3. What are the applications of vector-valued function ellipses?

Vector-valued function ellipses have various applications in mathematics and physics. They are used to describe the motion of objects in space, such as planets orbiting around a star. They are also used in engineering to model the movement of mechanical systems.

## 4. How can one graph a vector-valued function ellipse?

To graph a vector-valued function ellipse, one can use a computer program or graphing calculator that supports vector-valued functions. The function can be entered into the program, and the resulting graph will show the shape of the ellipse in the two-dimensional space.

## 5. Is it possible to have a vector-valued function ellipse in three-dimensional space?

Yes, it is possible to have a vector-valued function ellipse in three-dimensional space. In this case, the ellipse will be defined by a vector-valued function with three input parameters. The resulting shape will be a three-dimensional ellipsoid, rather than a two-dimensional ellipse.

• Calculus and Beyond Homework Help
Replies
3
Views
505
• Calculus and Beyond Homework Help
Replies
5
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
874
• Calculus and Beyond Homework Help
Replies
11
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
2K