Vector Functions -Write a in the form a=a_T T + aN_N

Question

Write a in the form

a=a_T T + aN_N

at the given value of t without finding T and

N.

r(t) = (5t^2)i + (5t+5/3t^3)j + (5t-5/3t^3)k, t=0

I need to fill T and N

a(0) = (?)T + (?)N

Answer 1

aT(t) = (r'(t)⋅r"(t))/|r'(t)|

aN(t) = |r'(t)χr"(t)|/|r'(t)|

The ⋅ means dot product. The χ means cross product. The absolute value lines || represent the length of the vector inside them. Once you calculate aT and aN then plug in 0 and you should get your answer. I'd show you the steps but its a lot of steps and hopefully you should be able to do this on your own if you know how to handle derivatives, dot products, and cross products of vector functions. If not, then I can help you out in an online tutoring session.