Question

(a) Find the unit tangent vector, T(t) and the unit normal vector, N(t), for the space curve r(t) cos(4t), sin(4t), 3t >. (b) From part (a), show that T(t) and N(t) are orthogonal

Answer 1

vettor and First of for mula of unit tangent au unit norm al vector ' (t) vector Unit tngeut Temam ber Unit normal vector T' (t) TIT IE) = coSAt, Sin4t), 3t > -4 sin (4t), 4 OS (4-t), 3 > -4sinat(4os (4 )) t9 16 sin24t+I6 cos4t +9 O Y vectoristhe n tongent (t) unit The 4 sinat), 4 COS4t, require norma ve cto will ow he unt ta The unit the denvative of magnitu tangent and Sin4t, o 16 s4t ) T'Lt) So i612 2dttsin2 4 16 AY

16 16 sin4t CDS4t 14s 16/5 os(4t),sin 4t, 0 O Y Means Ortho gonal Dot product o) TIE NI= LoS4 t, sinat, o) /-4 sina t), 4 os 4t, So 4 sin4twst 4 wS 4t. Sin 4t orthogonal Cre Ti and N() So