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Trick here is to write the velocities in component form.
Let (3,v_1) be the velocity of first particle.
Let (-4,v_2) be the velocity of second particle.

Since the vertical component of initial velocity is 0 for both the particles, we have v_1=v_2=gt=10t

We want to know the time at which the velocities are perpendicular, so simply set the dot product of velocities equal to 0  and solve t

(3,10t)\cdot~(-4,10t)=0\implies -12+100t^2=0\implies~t=\sqrt{0.12}s