Vector-Valued Function Ellipse

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.
  • #1
vandersmissen
19
0

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


Homework Equations



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.
 
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  • #2
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.
 

Related to Vector-Valued Function Ellipse

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.

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