# solve quickly

Let i^ and j^ be unit vector in horizontal and vertical respectively.

Let velocity of rain Vr=ai^+bj^

speed of rain = √(a^2+b^2)

Case I:

When relative velocity of rain w.r.t man is vertical

Vrm = Vr−Vm

Vm = 4i^

Vrm = (a−4)i^+ bj^

Since Vrm is vertical

a−4=0 => a=4

Case II:

When relative velocity is at 45∘

Vm = 8i^

Vrm=(a−8)i^+bj^

=−4i^+bj^

Since tanθ = b/(−4)

tan 45=1

=> |b|=4

Therefore, Speed = √(4^2+4^2) =4√2

Hence, the actual velocity of rain is 4√2 m/s