From vector function to xy function?

Question

Hi.
I am wondering how one finds the equation in x and y from a vector function and vice versa? It would be good if you could explain from these 2 examples:
1) r(t) = (cos t)i + (sin t) j + tk
2) r(t) = (t+1)i + (t^2 -1)k
Please show your work.
Thank you.

mecdub · Accepted Answer

1) r(t) = (cos t)i + (sin t) j + tk

The domain of all 3 components is all t, so we don't have to restrict t.

Parameterize the equation. 

x = cos t
y = sin t
z = t

Eliminate the parameter in the first two equations.(Get rid of the t)

sin² t + cos² t  = 1

x² + y² = 1

In the x-y plane, this is a circle or radius 1, centered at the origin.
z is not constant, so in 3-D space, this is a helix, or spring, centered at the origin, that circles the z axis.

2) r(t) = (t+1)i + (t^2 -1)k

Again, the domain is all reals.

Parameterize
x = t + 1
y = 0
z = t² - 1

Eliminate the parameter.

z = (x - 1)² - 1

Clearly a parabola in the x-z plane with the vertex at (1,0,-1).

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From vector function to xy function?

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