The sum of the magnitudes of two forces acting at a point is 18 and the magnitude of their resultant is 12. If the resultant is at `90^(@)` with the force of smaller magnitude, What are the magnitudes of forces?

Explanation

Let P be the smallar force, them it is given that
`P + Q = 18" "……(i)`
` R = [P^(2) + + Q^(2) + 2 PQ cos theta ]^(1//2) = 12" " ……..(ii)`
and `Q sin theta//[P +Q cos theta ] = tan phi = tan 90^(@) = oo`
`i.e.," " P + Q cos theta = 0" "........(iii)`
Substituting the value of P from Eqn. (i) in (iii),
`" "Q(1-cos theta ) = 18" ".......(iv)`
Now subtracting square of Eqn. (ii) from (i) ,
`" "2PQ(1 cos theta) = 18^(2) - 12^(2) = 180" ".....(v)`
Dividing Eqn. (v) by (iv),
`2 P= 10 , " " i.e., P=5`
Substitting for P in Eqn. (i),
`" "Q= 13`
So, the magnitudes of forces are 5 and 13 units.