Chapter 12.3, Problem 55E

(a)

To determine

To calculate: The value of $v\left(t\right)$, $\Vert v\left(t\right)\Vert$ and $a\left(t\right)$, where a particle moves on an elliptical path given by the vector-valued function $r\left(t\right)=6\cos ti+3\sin tj$.

(b)

To determine

To fill: The given table with the help of graphing utility, where a particle moves on an elliptical path given by the vector-valued function $r\left(t\right)=6\cos ti+3\sin tj$.

t	0	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\frac{2\pi }{3}$	$\pi$
Speed

(c)

To determine

To graph: The elliptical path which is given by the vector-valued function $r\left(t\right)=6\cos ti+3\sin tj$ and the velocity and acceleration vectors at the values of t given in the table in part (b).

(d)

To determine

The geometric relationship between the velocity and acceleration vectors, when the speed of the particle is increasing and when it is decreasing, where a particle moves on an elliptical path given by the vector-valued function $r\left(t\right)=6\cos ti+3\sin tj$.