The resultant of `vec(P)` and `vec(Q)` is `vec(R)`. If `vec(Q)` is doubled, `vec(R)` is doubled, when `vec(Q)` is reversed, `vec(R)` is again doubled. Find P:Q: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)+4PQ cos 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)+2PQ cos (180^(@)-theta)`
This yields `4R^(2)=P^(2)+Q^(2)-2PQ cos theta`..(iii)
Solving equation (i),(ii) and (iii) we obtain `Q=sqrt(3/2)R` and P=R`.