CHAPTER 1 VECTOR SCALARS AND Example 1.13 Find the equation of a straight line that passes through two given points A and B having position vectors and with respect to an origin O. a b A r O B b a P Let r be the position vector of any point P on the line through A and B. Then,   OA AB OB OA  AP  OP ( ;   But r t t b Thus .e .  i t t AB  ) a AP a OP   a 0 If the equation is written  , the sum of the coefficients of A t t b r , and is 1-t+t-1=0. Alternatively, since and are collinear, we have a b r AP PB m n for some scalars and m AP  n PB ( ) ( )  or m r a n b r  m a n b so that  r  m n which we call the symmetric form. Example 1.14 4 8 At noon, two boats P and Q are at points whose position vectors are  and iˆ ˆj 4  3 respectively. Both boats are moving at constant velocity; the velocity of P is iˆ ˆj 4 2 5 iˆ  and the velocity of Q is iˆ  (all distances are in kilometers and the ˆj ˆj
CHAPTER 1 VECTOR AND SCALARS Example 1.13 Find the equation of a straight line that passes through two given points A and B having position vectors a and b with respect to an origin O. A O a r b P B Let r be the position vector of any point P on the line through A and B. Then, OA OA   AB  AP  OB OP But Thus i .e . AP OP r    a t   t ( b t AB a ) ; t If the equation is written  A t a t b r  0 , the sum of the coefficients of a , b and r is 1-t+t-1=0. Alternatively, since AP and PB are collinear, we have for some scalars m and n m AP  n PB m a  n b r  m  n or so that m ( r a )  n ( b r ) which we call the symmetric form. Example 1.14 At noon, two boats P and Q are at points whose position vectors are 4 iˆ  8 ˆj and 4 iˆ  3 ˆj respectively. Both boats are moving at constant velocity; the velocity of P is 4 iˆ  ˆj and the velocity of Q is 2 iˆ  5 ˆj (all distances are in kilometers and the