Exercise 22

A standard parameterization for the unit circle is $\langle \cos(t), \sin(t) \rangle$, for $0 \le t \le 2\pi$.

1. Find a vector-valued function $\vr$ that describes a point traveling along the unit circle so that at time $t=0$ the point is at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$ and travels clockwise along the circle as $t$ increases.

2. Find a vector-valued function $\vr$ that describes a point traveling along the unit circle so that at time $t=0$ the point is at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$ and travels counter-clockwise along the circle as $t$ increases.

3. Find a vector-valued function $\vr$ that describes a point traveling along the unit circle so that at time $t=0$ the point is at $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$ and travels clockwise along the circle as $t$ increases.

4. Find a vector-valued function $\vr$ that describes a point traveling along the unit circle so that at time $t=0$ the point is at $(0,1)$ and makes one complete revolution around the circle in the counter-clockwise direction on the interval $[0,\pi]$.