Parametric representation of a curve

Question

Question:

Parametric Representation of a Curve

1. Find a parametric representation of the curve:

                            x² + y² = 36 and z = 1/∏ arctan (x/y).

i.e find a representation in the form: x = x(t); y=y(t), z = z(t)

2. Find what kind of curves are given by the following representations and draw (schematically) the curves:
                              i) r(t) = (2t - 5, -3t + 1,4):
                              ii) r(t) = (0, -cos(t), 3sin(t))

3. Find the equations of the tangent (straight line) and the normal plane to the curve given by the followig parametric representation:

                               x = 2 + 3t⁴; y = 2t + t³, z = t at the point t = 1

4. The equations of motion of a point particle in space are given by:

                              i) x(t) = 2t³ - 3; y(t) = -3t³; z(t) = 4t³ - 1
                              ii) x(t) = asin(wt); y(t) = acos(wt); z(t) = t

In each case, calculate the velocity vector v and the acceleration vector a. Also calculate the absolute values of these vectors. What are the curves along which the particle moves? (find these values for an arbitrary t, with a and w being fixed parameters).