Question

Given ř(t) =< 2 cost, t, 2 sint > as a trace of a moving object. (a) Find the curvature of K(t). (b) Find the arc length when 0<t <31. (c) Find the unit normal and binormal vectors of F(t).

Answer 1

Then pct) = {400gt, t, 2 sine) R!(t) = (-2sint, 1, 2 wst) Jall (t) = (-2east, 0, -2 sinty 1177'60) = \fasin+)?+ (0)2+(aust)? 11 foow 4+1 11 15 11013 = 515 a, K NOW, V (t) x 8" (4) 2l0st -zsint -zost o -2sint a (-asint) i - (45197 +420$t). +(2003+) (-asint, -4 2.10st) ll pic mp3" (i = (-a sint)? + (9*(Quast)? = 215 ITI(49 pct) 11 Finally the curvature Kit)= || (+)1² 215 555 Are length= fost une at Sole ve Jo dt STO T5

Cunit tangent rector 7(+) = Filca Ini(1) (pastast, tas ng cost) The unit normal veeton N(t) = 7(H) +7(4) 710) - tost 0 - 3 sind 1177 C+) con must) 70+ ( sint)? NCH) = (- COS t, o, sent) (-cost, o -sint) And the unit Binormal vector B(t)= 7 (4) X.NCA) k posmnt to seest - Sint -cost o

Vse sint. I - ( 7 sent + 2 1 3. Lost i + / cost {-5677, -2, cesty 15