The sum of the magnitudes of two vectors is 18 and the magnitude of their resultant is 12. If the resultant is perpendicular to one of the vectors, then what are the magnitudes of the two vectors ?

P+Q=18…..(1)
and ${\mathrm{P}}^2+{\mathrm{Q}}^2+2\mathrm{PQcos}\mathrm{\theta }=(12{)}^2=144.....(2)$
Further $\frac{\mathrm{Qsin}\mathrm{\theta }}{\mathrm{P}+\mathrm{Qcos}\mathrm{\theta }}=\tan {90}^{\circ }=\mathrm{\infty }$
or $\mathrm{P}+\mathrm{Qcos}\mathrm{\theta }=0....\left(3\right)$
From eq.(2),P² +Q² -2P x P =144
or Q²-P² =744
or (Q+P)(Q-P) =144
or (Q -P) =8 ...(4)
Adding eqs. (1) and (4), we have 2 Q =26 or Q = 13,
Now P =5
Hence, the magnitudes are 5 and 13.