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

`R^(2) = P^(2)+Q^(2) +2PQ cos theta " "..(i)`
When Q is doubled ,R is doubled
`4 R^(2) = P^(2) + 4 Q^(2) + 4 PQ cos theta " "…(ii)`
Also , `4 R^(2) = P^(2) +Q^(2) - 2Pq cos theta " "…(iii)`
Multiplying equation (i) be 2 ,
`2 R^(2) = 2P^(2) +2Q^(2) + 4PQ cos theta " "...(iv)`
Subtracting equation (iv) from equation (ii)
`2R^(2) = 2Q^(2) - P^(2) " "(v)`
Multiplying equation (iii) by 2 adding to equation (ii)
`12 R^(2) = 3P^(2) + 6 Q^(2) " "...(vi)`
From equations (v) and (iv)
`sqrt(3) R = sqrt(2) Q = sqrt(3)P `
`P : Q , R = sqrt(2) " sqrt(3) : sqrt(2)`