Calc 3 Problem

[caffeine206]

Major Tom is on a malfunctioning rocket is that is traveling according to the path
r(t) = (e^2t; 3t^3; t^2 + 10)
After contacting ground control, he hopes to reach a repair station located at the point (25e^4; 456; 62), where
t is measured in minutes and spatial coordinates are measure in miles. At t = 2 the rocket’s engines suddenly
cease, and he loses communication with ground control. Will major Tom make it to the repair station? (After the
rocket’s engines stop working it will follow the path of the tangent line to its path at that point.)

So far, I have taken the derivative of r(t) to get the velocity, v(t) = (2e^2t, 9t^2, 2t)
and plugging in t=2, I get (2e^4, 36, 4)

What's next?

 

[MarkFL]

Once the engines have stopped working, then Maj. Tom's velocity is constant (assuming no gravity from a nearby body) as given by the vector you found. Henceforth, find his new linear position vector as a function of time, and see if the coordinates of the repair station are on this line.