CHAPTER 3 DIFFERENTIATION VECTORS OF d B  2iˆ kˆ dt c) dt 2 0  A iˆ kˆ ˆj  t t 2 d B d t 1   d B A  6 2k     ˆ ˆ dt dt UNIT TANGENT VECTORS:   is If is a position vector in space, then the direction vector denoting OP At d dt A P. P parallel to the tangent to the curve at The unit tangent vector at can be found T At d  from  Tˆ dt At d  dt T P O  At Example 3.5 Find a unit tangent vector to any point on the curve where a, b, w are constants. The vector equation of the  cos  sin  x a wt, y a wt, z bt curve is  cos  sin  r a wtiˆ a wtˆj btkˆ d r sin cos    wtiˆ aw wtˆj bkˆ dt d r ds    a w2 b 2 2 dt dt
CHAPTER 3 DIFFERENTIATION OF VECTORS c) d B dt  2iˆ kˆ A  d B dt iˆ  t 2 2 ˆj t 0 kˆ d dt A  d B dt t 1     ˆ  6 ˆ  2k UNIT TANGENT VECTORS: If OP is a position vector At  in space, then the direction vector denoting d dt  is parallel to the tangent to the curve at P. The unit tangent vector T at P can be found d from Tˆ  dt d dt At  At  T O At  P Example 3.5 Find a unit tangent vector to any point on the curve x  a cos wt, y  a sin wt, z  bt where a, b, w are constants. The vector equation of the curve is r  a cos wtiˆ  a sin wtˆj  btkˆ d r dt  sin wtiˆ  aw cos wtˆj  bkˆ d r dt  ds dt  a 2 w2  b 2 A