Question

Referred to a fixed origin O, the position vectors of the points P and Q are (6i-5j) and (10i+3j) respectively. The midpoint of PQ is R.

a) Find the position vector of R.

b) The midpoint of OP is S. Prove that, SR is parallel to OQ.

Answer 1

position vector of P = 6i - 5j 
e.g. vector OP = 6i - 5j 
in co-ordinate system point P ≡(6,-5)
similarly ,
position vector of Q = 10i + 3j 
e.g. vector OQ = 10i + 3j 
in co-ordinate system point Q ≡ (10,3)

O(0,0)_______________P(6,-5)_______R_________Q(10,3)
a/c to question,
R is the midpoint of PQ 
we know formulae of midpoint in co-ordinate geometry  use here,
let R ≡(x,y)
then, x = (6+8)/2 = 8
         y = (-5+3)/2 = -1
hence point R ≡(8, -1)
hence, position vector  R = 8i -j 



(b) the midpoint of OP is S.
SO, point S≡(6/2,-5/2)=(3,-2.5)
O(0,0)__________S(3,-2.5)__________P(6,-5)_____R(8,-1)_______Q(10,3)

vector SR = position vector OR-position vector OS 
                 =8i - j - (3i -2.5j) 
                 = 5i +1.5j 

again,
vector OQ = 10i +3j = 2{5i + 3/2j} = 2{5i+ 1.5j}

here, it is clear that,
     vector SR = (1/2)vector OQ
hence, vector SR is parallal  to OQ