CHAPTER 3 DIFFERENTIATION VECTORS OF d r  dt ; ds 1 1 ds dt r 6     dt r r r ds  6 r 1 r r r r r  2    r r 6    2  6 2 r r r   Thus  r    6   r     r QUESTIONS . k 1. Find and for the space curve   Tˆ sin , cos , 4 sin2 x  y  z    cos , K  cos2  cos 2 sin sin2 Ans. 12 cos 4 6 3  1 2 T  8 ,  z t ,  y .  2 2. Find the torsion of the curve  x our t 2 2t 1 t t T=0. x-3y+3z=5 Ans. The curve lies on the plane 3. Show that the equation of the tangent line, principal normal and bi-normal to the space curve at the point t=to can be written respectively  r rt t where is a parameter. , , ,       r r t Tˆo r r t Nˆ r r t Bˆo o o o o 4. Find equations for (a) tangent, (b) principal normal and (c) bi-normal to the
CHAPTER 3 DIFFERENTIATION OF VECTORS ds 6    r r r  6 r ; dt ds  1 ds dt  1 d r dt 1 r Thus      r r r  6 2     r r   2  6 r r r  6  r    r r   2 r r QUESTIONS 1. Find Tˆ and k for the space curve x   sin  , y  cos , z  4 sin2 . Ans. K  1 8 6 2 cos , T  3  cos cos2  2 sin  sin2  12 cos 4 2. Find the torsion of the curve x  2t 1 t , y  t 2 t , z  t  2 . Ans. T=0. The curve lies on the plane x-3y+3z=5 3. Show that the equation of the tangent line, principal normal and bi-normal to the space curve r  rt at the point t=to can be written respectively r  r o  t Tˆo , r  r o  t Nˆ o , r  r o  t Bˆo , where t is a parameter. 4. Find equations for (a) tangent, (b) principal normal and (c) bi-normal to the our