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 P2+Q2+2PQcos⁡θ=(12)2=144.....(2)Further Qsin⁡θP+Qcos⁡θ=tan⁡90∘=∞or P+Qcos⁡θ=0....(3)From eq.(2),P2 +Q2 -2P x P =144or Q2-P2 =744or (Q+P)(Q-P) =144or (Q -P) =8 ...(4)Adding eqs. (1) and (4), we have 2 Q =26 or Q = 13,Now P =5Hence, the magnitudes are 5 and 13.