Answer:
graph the equation (see attached)
Explanation:
You want to find the path of a particle defined by the vector equation ...
r = 4cos(3t)i +6sin(3t)j
Path
When t=0, the position is ...
r = 4(1)i +6(0)j = 4i
When t=π/6, the position is ...
r = 4(0)i +6(1)j = 6j
You can continue to evaluate r for different values of t, and you will find the points form an ellipse centered at the origin. Its major axis extends from y=-6 to y=6, and its minor axis extends from x=-4 to x=4. The graph is attached. Motion is counterclockwise from the +x axis.
It works nicely to let a graphing calculator plot the points.