Scaling the edge length of a graph to be equal to edge weight (again)

Question

This is a follow up to my previous question posted here

The following is an excerpt of the solution provided here to set the edge length of a graph equal to the edge weights

edges = {1 <-> 2, 1 <-> 3, 1 <-> 4, 2 <-> 5, 2 <-> 6, 5 <-> 6, 
   3 <-> 4, 3 <-> 7, 6 <-> 7, 7 <-> 8, 2 <-> 9};

vd = {{75., 25., 0}, {115., 45., 0}, {10., 5., 0}, {45., 0, 0}, 
  {90., 60., 0}, {45., 55., 0}, {0, 25., 0}, {10., 50., 0}, {115.,  25.,0}};

vl = Range[Length@vd];

vcoords = MapIndexed[#2[[1]] -> # &, vd];
ew = {1 \[UndirectedEdge] 2 -> 49.6, 1 \[UndirectedEdge] 3 -> 74.4, 
 1 \[UndirectedEdge] 4 -> 49.6, 2 \[UndirectedEdge] 5 -> 37.2, 
 2 \[UndirectedEdge] 6 -> 74.4, 5 \[UndirectedEdge] 6 -> 49.6, 
 3 \[UndirectedEdge] 4 -> 37.2, 3 \[UndirectedEdge] 7 -> 24.8, 
 6 \[UndirectedEdge] 7 -> 62, 7 \[UndirectedEdge] 8 -> 37.2, 
 2 \[UndirectedEdge] 9 -> 24.8}

g3d = Graph3D[vl, edges, VertexCoordinates -> vcoords, 
  EdgeWeight -> ew, VertexLabels -> Placed["Name", Center], 
  EdgeLabels -> {e_ :> Placed["EdgeWeight", Center]}, 
  VertexSize -> .3, VertexStyle -> Red]
vars3d = Array[Through[{x, y, z}@#] &, Length @ vd];

λ = 1/100.;

obj3d = Total[(Norm[vars3d[[First@#]] - vars3d[[Last@#]]] - # /. ew)^2 & /@ 
  EdgeList[g3d]] +  λ Total[Norm /@ (vars3d - vd)];

lbnd = 0;
ubnd = 500;

solution3d = Last@Minimize[{obj3d, And @@ Thread[lbnd <= Join @@ vars3d <= ubnd]}, 
    Join @@ vars3d];

edgeLengths3d = # -> Norm[vars3d[[First@#]] - vars3d[[Last@#]]] /. 
     solution3d & /@ EdgeList[g3d];

Grid[Prepend[{#, # /. ew, # /. edgeLengths3d} & /@ 
   EdgeList[g3d], {"edge", "EdgeWeight", "Edge Length"}], 
 Dividers -> All]

I would like to know how to modify the above when the nodes of the graph aren't numbered consecutively.

I tried the following,

edges = {1 <-> 2, 1 <-> 3, 1 <-> 4, 2 <-> 11, 2 <-> 6, 11 <-> 6, 
   3 <-> 4, 3 <-> 7, 6 <-> 7, 7 <-> 8, 2 <-> 10};
vd = {{75., 25., 0}, {115., 45., 0}, {10., 5., 0}, {45., 0, 0}, {90., 
    60., 0}, {45., 55., 0}, {0, 25., 0}, {10., 50., 0}, {115., 25., 
    0}};

vl = {1, 2, 3, 4, 11, 6, 7, 8, 10};

vcoords = Thread[vl -> vd];

ew = {1 \[UndirectedEdge] 2 -> 49.6`, 1 \[UndirectedEdge] 3 -> 74.4`, 
  1 \[UndirectedEdge] 4 -> 49.6`, 2 \[UndirectedEdge] 11 -> 37.2`, 
  2 \[UndirectedEdge] 6 -> 74.4`, 11 \[UndirectedEdge] 6 -> 49.6`, 
  3 \[UndirectedEdge] 4 -> 37.2`, 3 \[UndirectedEdge] 7 -> 24.8`, 
  6 \[UndirectedEdge] 7 -> 62, 7 \[UndirectedEdge] 8 -> 37.2`, 
  2 \[UndirectedEdge] 10 -> 24.8`};

g3d = Graph3D[vl, edges, VertexCoordinates -> vcoords, 
  EdgeWeight -> ew, VertexLabels -> Placed["Name", Center], 
  EdgeLabels -> {e_ :> Placed["EdgeWeight", Center]}, 
  VertexSize -> .3, VertexStyle -> Red]

vars3d = Array[Through@{x, y, z}@vl[[#]] &, Length@vl];

λ = 1/100.;

obj3d = Total[(Norm[vars3d[[First@#]] - vars3d[[Last@#]]] - # /. 
         ew)^2 & /@ EdgeList[g3d]] + \[Lambda] Total[
     Norm /@ (vars3d - vd)];

I couldn't execute obj3d successfully, the following is displayed

Part::partw: Part 11 of {{x[1],y[1],z[1]},{x[2],y[2],z[2]},{x[3],y[3],z[3]},{x[4],y[4],z[4]},{x[11],y[11],z[11]},{x[6],y[6],z[6]},{x[7],y[7],z[7]},{x[8],y[8],z[8]},{x[10],y[10],z[10]}} does not exist.

Suggestions on how to modify the expression for obj3d and the lines below it to successfully use the solution provided in the previous post for the new input will be highly appreciated.

Notebook

EDIT: The suggestion provided below resolved the error reported above. Next, I did the same modification for line

edgeLengths3d = # -> 
Norm[[Through@{x, y, z}@First[#] -  Through@{x, y, z}@Last[#]]] /. solution3d & /@ EdgeList[g3d];

and the following error occurs

Part::pkspec1: The expression {x[1]-x[2],y[1]-y[2],z[1]-z[2]} cannot be used as a part specification.

Could you please suggest how this line has to be modified?

Answer 1

Change the definition of obj3d to

obj3dB = Total[(Norm[Through @ {x, y, z} @ First[#] - 
           Through@{x, y, z} @ Last[#]] - # /. ew)^2 & /@ EdgeList[g3d]] + 
    λ Total[Norm /@ (vars3d - vd)]