The sum of the magnitudes of two forces acting at a point is 18 and the magnitude of their resultant is 12. If the resultant is at `90^(@)` with the force of smaller magnitude, What are the magnitudes of forces?

Explanation

Let ` vec A and vec B` be the two forces and theta be the angle betweem them . Let A lt B.
As per question, ` A + B =18`
or ` B= (18 -A)
As the resultant makes an anle `90^(@)` with the force of smaller magnitude , then ` beta = 90^@`.
` tan beta = (B sin theta) /(A+ B cos theta) or tan 90^@ = (B sin theta) /(A +B cos theta)
or ` prop = ( B sin theta0 /(A + B cos theta) or A + B cos theta =0`
or ` B cos theta =- A` ..(i)
Resultant , ` R= sqrt( A^2 + B^2 + AB cos theta )`
(12)^2 = A^2 + (s8 - A0 ^2 + 2 A (- A)`
[ From (i) and (ii) ]`
or ` 144 = A^2 + 324 - 36 A + A^2 -2 A^2`
or ` 36 A 180 or A = 5 N , B=18 -5 = 13 N`.