Parametric Equations

[DeadxBunny]

Original question:

a) Say r'(t) = 3t^2 i - cost j + 2t k, and r(0) = i + k. Find r(t).
b) Find T(t).
c) Find parametric equations for the tangent line to the curve at t=1.

I have done parts a and b and got the following results:

a) r(t) = t^3 + 1 i - sint j + t^2 + 1 k
b)T(t) = r'(t)/|r'(t)| = (3t^2 i - cost j + 2t k)/(sqrt(9t^4 + cos^2(t) + 4t^2))

Are these answers correct so far? I'm unsure about my answer for T(t) because the denominator seems so messy.

Also, and most importantly, how would I do part (c)?

Thanks!!

[vsage]

Multivariable calc is messy. I wouldn't worry about your denominator because it seems right to me. T(t) gives you a tangent vector to a function at any parameter t right? Now you know that the slope for x, y and z by the i, j and k components of T'(t), respectively. Now all you need to do is offset the slopes by constants which can be found from r(0) in the original equation.