Parametric Equations Homework - Find Curve & Position Vector

[chrsr34]

Homework Statement

Consider the curve of intersection of the cylinders [x^2+y^2=4] and [z+x^2=4]. Find parametric equations for this curve and use them to write a position vector.

Homework Equations

Thats what I am looking for. What to set t equal to.

The Attempt at a Solution

I set t=x and got a square root for y. So if i set t=x^2, i get rid of the square root for y, but I am not sure if this is correct. i really don't know the rules for parametric equations. Any input is appreciated.

Chris

 

[Dick]

An easy choice for t is the angular coordinate in the x,y plane. So x=2*cos(t), y=2*sin(t). Given this, can you figure out what z is in terms of t? Unfortunately, I don't think there are any general rules for doing this. You just have to make a picture in your mind of what the curve looks like and then look for a choice for t.

 

[chrsr34]

I see. Yes i could find z with those coordinates. But is there a reason you chose those? Are those cylindrical coordinates? (im a little rusty).
Is there anything wrong with setting t=x^2?

 

[Dick]

The x^2+y^2=4 cylinder intersects the x,y plane in a circle. It's easy to parametrize that in polar coordinates. If you take t=x^2 then you still have a square root for y (contrary to what you said), so you'd have to define the curve as a union of pieces.

 

[chrsr34]

All yes your right.
So my position vector should be: r = < 4cos^2(t), 4sin^2(t), 4-4cos^2(t) >
Is this correct?

 

[Dick]

Noooo. x=2*cos(t), not 4cos^2(t)!

 

[chrsr34]

yea i just realized i did that.
So, r = < 2cos(t), 2sin(t), 4-4cos^2(t) >

This look good?