The sum of the magnitude of two forces acting at a point is 18 and the magnitude of the resultant is 12.if the resultant is at 90 degree with the force of the smaller magnitude,what are the magnitudes of forces.

Let P and Q be the forces and P is greater than Q, acting at a point O inclined at some angle with each other.

Here, P+Q = 18 ....(1)

and the resultant  R=12

Construct a parallelogram with sides with OA = Q, OB = P, with angle AOB as an obtuse angle and OA along the horizontal direction.

Draw the diagonal OC  to the parellelogram OACB  so that angle AOC =90^o .

OC is the resultant  R which is perpendicular to the smaller force Q.  

 

Consider triangle AOC.

OA = Q

AC = OB  = P

and angle AOC =90^o

 

Therefore, triangle AOC is a right angled triangle with AC = P as hypotenuse.

 

Then   P²  = Q² + R² 

or, P² - Q² = R² 

or, (P+Q) (P - Q)  = R²  

or, P -Q  = R²/P+Q

or, P- Q  = 144/18 = 8  ...(2)  

 

 

Adding (1) and (2)  

 

2P = 26

or, P = 13

Then Q = 18-13=5