Question

the sum of magnitudes of 2 forces acting at a point is 18 and magnitude of their resultant is 12.If the resultant is perpendicular with force of smaller magnitude what are magnitudes of forces?
(ii)p+q=r(they are in vector form)p=q=r/root2
then find the angle between pair of vectors p,q and q,r and r,p

Answer 1

Let $\vec{P}\ and\ Q$ be the two vectors.  Let Q < P.
Let the angle between the two vectors be Ф.

Resultant of $\vec{P}\ and\ \vec{Q}:\ \ \vec{R}=\vec{P}+\vec{Q}$

     R = P² + Q² + 2 P Q Cos Ф  --- (1)

   We are given P + Q = 18, => Q = 18 - P.             and,      R = 12.
   Angle between R and Q = 90°.

We have  $\vec{P}=\vec{R}-\vec{Q}$ .
So    P² = R² + Q² - 2 R Q Cos 90°    ----- (2)
            =  R² + Q² - 0
             = 12² + (18 - P)²
             = 144 + 324 + P² - 36 P

  =>  36 P =  468
  =>  P = 13
  =>  Q = 18 - 13 = 5
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Given R is perpendicular to Q. so angle is 90 deg.
Use the equation (1) to find the angle Ф between P & Q, as magnitudes of P, Q and R are known. 

 The angle between  R and P could be equal to (Ф - 90) deg.