For the curve given by rt Find the unit tangent Tt Find th

For the curve given by r(t) = Find the unit tangent T(t) = Find the unit normal N(t) Find the curvature k(t)

Solution

r\'(t) = <-4cost, 1, -4sint>, |r\'(t)| = [(4cost)² + 1 + (4sint)²] = 17

a) T(t) = r\'(t)/|r\'(t)| = <-4cost/17, 117, -4sint/17>

b) dT/dt = <-4sint, 0, -4cost>/17

|dT/dt| = 4/17

N(t) = (dT/dt)/|dT/dt| = <-4sint, 0, -4cost>/4 = <-sint, 0, -cost>

c) k = |dT/dt|/|r\'(t)| = (4/17)/17 = 4/17