A motion in R^3 is given by r(t)=<2t,sin2t,cos2t>, t >=0. This motion applies to both questions?

Question

1.) Name the Trajectory?

2.) How to compute the unit vectos T and N at the instant t= pi/4

Answer 1

The trajectory is a helix.

v(t) = r'(t) = (2, 2 cos 2t, -2 sin 2t)
a(t) = r"(t) = (0, -4 sin 2t, -4 cos 2t).
v(π/4) = (2, 0, -2)
a(π/4) = (0, -4, 0).

Now T is a unit vector in the direction of v(t), so it will be v(t) / ||v(t)||, i.e.
T = (2, 0, -2) / (2√2) = (1/√2, 0, -1/√2)

Now we write a = a_t T + a_n N where T.N = 0. To find N, we compute a_t and then take N to be a normalised version of a - a_t T.

Computing a_t is simple: a_t = a.T (to see this, a.T = a_t T.T + a_n N.T = a_t (1) + a_n (0) = a_t). In this case a_t = (0, -4, 0) . (1/√2, 0, -1/√2) = 0.
So a_n N = a - a_t T = (0, -4, 0). So we let a_n = 4 and N = (0, -1, 0).