Exercises

1. Suppose that and . Compute the following quantities.
   1. 
   2. 

2. Consider the two-dimensional position vector , where b is a positive constant.
   1. Show that the trajectory, or image curve, of is the circle of radius 4 centered at the point . That is, show that the component functions and of satisfy the equation

      

      for all values of t. Explain why this means that the image curve is the circle in question.

   2. Show that the speed is constant and explain why this is so. Then find a value of the constant b that gives a speed of 20.
   3. Show that is perpendicular to .
3. Suppose particle 1 is going around a circle and its position is given by . Particle 2, on the other hand, is going back and forth on the line y=ax with position function , where a is a constant.
   1. If a=2, find the value of t, , for which the two particles collide.

   2. Is it possible to find a positive real number a for which the two particles do not collide for any value of t, ? Explain your answer.

4. For the following position vector, determine the value(s) of t, , for which the speed is a minimum.

   

   Graphing the speed will allow you to locate the minimum speed approximately, but you should use the techniques from earlier in calculus to find accurate values.