To a man walking at a rate of 3km/hr,the rain appears to fall vertically downward. When hre increase the speed to 6km/hr it appears to meat him at an angle of 30^o with the downward vertical. Find the real direction and speed of rain

Let i^i^ and j^j^ be unit vector in horizontal and vertical respectively.Let velocity of rain vr=ai^+bj^vr=ai^+bj^speed of rain=a2+b2−−−−−−√=a2+b2Case I:when relative velocity of rain w.r.t man is verticalvrm=v¯r−v¯mvrm=v¯r−v¯mvm=3i^vm=3i^vrm=(a−3)i^+bj^vrm=(a−3)i^+bj^Since vrmvrm is verticala−3=0=>a=3a−3=0=>a=3Case II:when relative velocity is at 45∘45∘v¯m=6i^v¯m=6i^vrm=(a−6)i^+bj^vrm=(a−6)i^+bj^=−3i^+bj^=−3i^+bj^Since tanθ=b−3tan⁡θ=b−3tan 45=1=> |b|=3