Question

The resultant of two vectors P and Q acting at a point is R. Show that if P is doubled, Q remains unaltered, then the resultant will be of the magnitude √2P²-Q²+2R²

Answer 1

R = √P²+Q²+2PQcosΘ   ( Θ is the angle between the vectors)
⇒ R² = P²+Q²+2PQcosΘ
if P is doubled & Q remains unaltered then resultant 
R'=√4P²+Q²+4PQcosΘ
   =√(2P²+2P²)+(2Q²-Q²)+4PQcosΘ
   =√2P²-Q²+2(P²+Q²+2PQcosΘ)
   =√2P²-Q²+2R²  (proved)

Answer 2

Let the angle between vectors P and Q be = Ф.

magnitude of the resultant = R,   Hence,   R² = P² + Q² + 2 P Q Cos Ф

Let  P' = 2 P and Q  as well as  the angle between them Ф  remains the same.  Let the resultant vector be R'.

   R' ²  =  (2 P)² + Q² + 2 (2 P) Q Cos Ф
          =  4 P² + Q² + 4 P Q Cos Ф
          =  2 (P² + Q² + 2 P Q Cos Ф) + 2 P² - Q²
           = 2 R² + 2 P² - Q²

  Hence, the resultant now is
     R' = √ [ 2 R² + 2 P² - Q² ]