Measure the distance of multiple spots to a several line segments

I have 2 populations of 3D points placed on opposite sides of an volume (3D). I want to draw an orthogonal line from each point from the top population, along the z-axis. Then find the spot in the lower population that is closest to the line. Then measure the distance between each of the points.

I would like to be able to vectorize this, as the number of spots could be very high.

I have used this code below to measure that distance from the first point in P to a the line Q1Q2. I can make a for loop, but that is not very efficient. An help would be appreciated.

Q1=vVertices1(1,:); %contains 2000points

Q2=vVertices1(1,:);

Q2(:,3)=Q2(:,3)+1000; %To generate orthogonal line

P=vVertices2(1,:); %contains 2000 points

dist = sqrt(norm(Q2-Q1)^2*norm(P-Q1)^2-dot(Q2-Q1,P-Q1)^2)/norm(Q2-Q1);