Given a position vector rt lt e2t cos t sin t2 gt find the

Question

1. Find the curvature of the curve

x = 3 cos (t), y = 3 sin (t), z = 4t.

at t = π/6 by calculating ||T'(s)||.

2. Given a position vector

r(t) =< e^2t, cos t, sin t² >

Find the velocity and acceleration vectors and the speed at time t.

3. Find the equation of the plane that passes through the three points

P = (2,0,0), Q = (4,2,0), R= (0,1,2).

4. Find the arc length of the curve given by

r(t) = a cos (ωt) i + a sin(ωt)j.

between t = 0 and t = π/3.