CHAPTER 1 VECTOR SCALARS AND x2 y2 2 z2 2 Magnitude of Which is the distance   x1  y1  z1 PQ PQ between P and Q. DIRECTION COSINES: The direction of a vector in three dimensions is determined by the angles that the vector makes with the three axes of reference. Z P b  X Y Let     r OP r aiˆ bˆj ckˆ cos cos Then a  r c r cos  cos    a r r a  cos  cos a r r b  a2  b2  c2  r 2 Also cos2  cos2  cos2     r r r r 2 2 2 2 cos2  cos2  cos2  1    cos , cos  , cos ; l  m  n  If    1 l m2 n2 2 Then Note: [l,m,n] written in square brackets are called the direction cosines of the vector OP and are the values of the cosines of the angles which the vector makes with the three axes of reference. So for the vector    r aiˆ bˆj ckˆ ; ;    and of course    l m n r a2 b2 c2 a r b r c r Example 1.5 Find the direction cosines [l,m,n] of the vector  3iˆ 2  6kˆ r ˆj Here a=3, b=-2, c=6; 9 4 36 7 r     , , l  m  n  37 67
CHAPTER 1 VECTOR AND SCALARS Magnitude of PQ  PQ  x2 x1  y2 y1 2  z2 z1 2 Which is the distance between P and Q. DIRECTION COSINES: The direction of a vector in three dimensions is determined by the angles that the vector makes with the three axes of reference. Z P Let X OP  r  aiˆ  bˆj  ckˆ  r b Y Then a b c r r r  cos a  r cos  cos  a  r cos   cos a  r cos Also If Then a2  b2  c2  r 2 r 2 cos2   r 2 cos2   r 2 cos2   r 2 cos2   cos2   cos2   1 l  cos , m  cos  , n  cos ; l 2  m2  n2  1 Note: [l,m,n] written in square brackets are called the direction cosines of the vector OP and are the values of the cosines of the angles which the vector makes with the three axes of reference. So for the vector r  aiˆ  bˆj  ckˆ l  a r ; m  b r ; n  c r and of course r  a2  b2  c2 Example 1.5 Find the direction cosines [l,m,n] of the vector r  3iˆ 2 ˆj  6kˆ Here a=3, b=-2, c=6; r  9  4  36  7 l  37 , m  , n  67