Calculus 3: Parametric curves, conic sections

(x^2 + y^2 = 1). Parameterize the curve such that t is in the domain $[0, 2\pi]$.

Sketch the graph for the equation $x \cdot y = 1$ and describe its properties.

Sketch the curve whose vector equation is $r(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k}$.

Given the vector-valued function $r(t) = \langle 4\cos(t), 4\sin(t), 3 \rangle$, determine the curve it describes in 3D space.

Given the vector-valued function $r(t) = \langle 0, 3\cos(t), 5\sin(t) \rangle$, describe the curve in 3D space.

Given the vector-valued function $r(t) = \langle 6\cos(t), 0, 6\sin(t) \rangle$, describe the curve in 3D space and explain the effect of a negative $y$-component like $-t$.