Find the unit tangent vector of the given curve rt 7 2ti

Find the unit tangent vector of the given curve.

r(t) = (7 - 2t)i + (2t - 4)j + (9 + t)k

Solution

r(t) = (7 - 2t)i + (2t - 4)j + (9 + t)k

derivative of r(t)

r\'(t) = -2i + 2j +k

magnitude of r\'(t) = 4 +4+1 =9 = 3

unit tangent vector T(t) = r\' (t)/|r\'(t)|

T(t) = -2i/3 + 2j/3 +k/3