Question

                                  →
In the triangle below OA =a and OB=b. M is mid-point of Ab and N is a point on Ob such that ON=1/3OB. AN and OM intersect at P
        

1. Express the following vectors in terms of a and b
   1. AB 
   2. OM   
   3. AN  
2. If OP=tOM and AP=sAN, express OP in two different ways hence find the values of t and s. 
3. State the ratio AN:NP

Answer 1

1.  
   1. AB b − a
   2. OM ½a + ½b
   3. AN ¹/₃b − a
2.  →
   OP =t(½a + ½b)
   →
   OP = a + s(¹/₃b − a)
   ½t = 1 − s ....(i)
   ½t = ¹/₃s ....(ii)
   ½t = 1 − ³/₂t
   ⁴/₂t = 1
    t ½=
     ¼ = ¹/₃s
       s=¾
3. AN:NP
      4:−1