The resultant of `vecP` and `vecQ` is `vec R`. If `vecQ` is doubled, `vecR` is doubled, when `vecQ` is reversed, `vecR` is again doubled, find P : Q : R,

Explanation

Let `theta` be the angle between `vecP` and `vecQ` . Then
`R^(2) = |vecP + vecQ| ^(2) = P^(2) + Q(2) + 2PQ cos theta" "....(i)`
If `vecQ` is double. That means, the magnitude of
`vecQ` and `vecP` is 2R.
`(2R)^(2) = P^(2) + (2Q)^(2) + 2P(2Q) cos theta`
This yields, `4R^(2) = P^(2) + (2Q)^(2) + 4PQ costheta" ".....(i)`
When `vecQ` is reversed, `vecR` is double. Hence, the magnitude of resultant of `vecP` and `(-vecQ)` is 2 R.
Then, `(2R)^(2) = P^(2) + Q^(2) + 2PQ cos (180^(@)- theta)`
This yields `4R^(2) = P^(2) + Q^(2) - 2PQ cos theta" ".....(iii)`
Eq.(i) - eq.(iii) yield ,PQ `cos 0 = - (3R^(2))/(4)" ".....(iv)`
Eq.(i) + eq. (iii) yelds , `P^(2) + Q^(2) = (5R^(2))/(2)" ".....(V)`
Eq. (ii) + eq.(iv) yields, `P^(2) + 4Q^(2) = 7R^(2)" ".....(vi)`
Solving eq.(v) and eq. (vi) we obtain `Q = sqrt((3)/(2)) R and P=R`.
Hence, `P:Q : R = sqrt(2) : sqrt(3) : sqrt(2):`