2P and P are two vectors inclined to each other at such an angle that if the 1 st vector is doubled , the value of the resultant becomes three times. What is the angle between the two vectors ?

Explanation

Let the initial resultant be R and the angle between the two vectors 2P and P be `alpha` .
`therefore R^2=(2P)^2+P^2+2*2P*Pcos alpha`
`or, R^2=5P^2+4P^2 cos alpha` …….(1)
In the second case , 1st vector =2P`xx`2 =4P and resultant=3R
`therefore (3R)^2 =(4P)^2+P^2+2*4P*P cos alpha`
or, `9R^2=17P^2+8P^2 cos alpha`
or, `R^2=(17)/(9)P^2+8/9P^2 cos alpha` .........(2)
From (1) and (2)
`5P^2+4P^2 cos alpha =(17)/9 P^2+8/9 P^2 cos alpha`
or, `(4-8/9) cos alpha =(17)/(9)-5 or, (28)/(9) cos alpha= -(28)/(9)`
or, `cos alpha =-1 = cos 180^@ therefore alpha= 180^@`.