CHAPTER 1 VECTOR SCALARS AND From the adjoining figure; where  ( )  , OD x OA x a where y is a scalar ( ) , OC  y OB  y b x is a OR  OD  OC But law of gram addition) (parallelo r or   r x a y b Which is the required expression. Similar treatment for three non-coplanar vectors , and results into the expression: a b c    r x a y b z c in three dimensional space where x, y and z are scalars. Example 1.12 Prove that the diagonals of a parallelogram bisect each other.      We have AC a b ~ ~ C a P ~ B A D ~ ;   AP x AC BP y BD  But AB  BP  AP a  y b a  x a  b ~ ~ ~ ~ Comparing coefficients of and of on left and right we find that   a b x y 1 P and is the midpoint of the diagonals. 2
CHAPTER 1 From the adjoining figure; VECTOR AND SCALARS OD  x ( OA )  x a , OC  y ( OB )  y b , where where x is a y is a scalar But OR  OD  OC (parallelo gram law of addition) or r  x a  y b Which is the required expression. Similar treatment for three non-coplanar vectors a , b and c results into the expression: r  x a  y b  z c in three dimensional space where x, y and z are scalars. Example 1.12 Prove that the diagonals of a parallelogram bisect each other. We have AC      a ~ B b ~ P a ~ C A D ~ But AP  x AC ; BP  y BD AB  BP  AP a  y b a  x a  b ~ ~ ~ ~  Comparing coefficients of a and of b on left and right we find that x  y  and P is the midpoint of the diagonals. 1 2 r