How to interpolate 2d data with constant spacing?

I try to figure out a way to linear interpolate my data. i am not quite sure how to tell him that for each step it should add the 10 new values at the end of the vector _lin. There is probably a faster way to do this but i am really lost and on the verge of a mentally breakdown since i cant figure out these kind of "simple" tasks. If anyone has an idea for an approach on how to manage this linear interpolation over the entire profil with constant spaceing i would be very grateful. i tried a little with the interp1 but i guess you also have to make a loop for over each data point.

if true
vz=[  1.5000e+003  1.5397e+003  1.5577e+003 ; 
      0.0000e+000  2.0862e+001  1.0443e+002  ]
n=3
for i=1:n-1
v_lin=linspace(vz(1,i),vz(1,i+1),10);   % It would be also great if one could interpolate the entire plot 
v_lin=[v_lin v_lin(i)];                 % Here i want the interpolated data to just add for each loop
z_lin=linspace(vz(2,i),vz(2,i+1),10);
z_lin=[z_lin z_lin(i)];

end

[Sebastian]

so now i got this far, which is kinda ok. is there a way to interpolate with interp2 so that the one row gets constant spacing. where the vz(2,:) is suposed to be interpolated to zz=1:1:zmax and the vz(1,:) gets interpolated. i also tried with the mashgrid input and entering the form vz3 = interp2 (X, Y, Z, XI, YI).

 if true
    for i=1:n+1;     %i added an extra end point by hand
    v_lin(i,:)=linspace(vz(1,i),vz(1,i+1),int);      
    z_lin(i,:)=linspace(vz(2,i),vz(2,i+1),int);      
  end   
v_lin2=[]; 
z_lin2=[];
for k=1:n+1    
    y=v_lin(k,:).';
    p=z_lin(k,:).';
    v_lin2=vertcat(v_lin2,y);
    z_lin2=vertcat(z_lin2,p);
end
v_lin2=v_lin2';
z_lin2=z_lin2';
vz2=[v_lin2 ; z_lin2]
  end