The resultant of two vector P and Q acting at a point inclined to each other at an angle `theta ` is R . If the magnitude of vector Q is doubled ,R is also doubled .if the vector Q is reversed in direction R is again doubled .Find the ratio between P , Q and R .

Explanation

Let `theta` be the angle between `vec(P)` and `vec(Q)`.Then
`R^(2)=|vec(p)+vec(Q)|^(2)=P^(2)+Q^(2)+2PQ cos theta`…..(i)
If `vec(Q)` is doubled, `vec(R)` is doubled. That means ,the magnitude of resultant of `2vec(Q)` and `vec(P)` is
`(2R)^(2)=P^(2)+(2Q)^(2)+2P(PQ)cos theta`
This Yields `4R^(2)=P^(2)+4Q^(2)+4PQcos theta`....(ii)
When `vec(Q)` is reversed,`vec(R)` is doubled. Hence, the magnitude of resultant of `vec(P)` and `(-vec(Q))`is `2R`.
Then `(2R)^(2)=P^(2)+Q^(2)+2PQcos(180^(@)-theta)`
This yield `4R^(2)=P^(2)+Q^(2)-2PQcos theta`....(iii)
(ii)-(i) yields `3Q^(2)+2PQcos theta=3R^(2)`....(iv)
(i)+(iii) yield `P^(2)+Q^(2)=(5R^(2))/(2)`...(v)
(iii)+(iv) yield `P^(2)+4Q^(2)=7R^(2)`....(vi)
Solving (v) and (vi), we obtain `Q=sqrt(3/2)R`and `P=R`.
Hence,P:Q:R=`sqrt(2):sqrt(3):sqrt(2)`.