Vector-Valued Function Ellipse

[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

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.

 

[HallsofIvy]

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.