How to draw a cube to facilitate subsequent vector operations?

Question

This question is a continuation of the previous one：

How to draw a cube?

Draw a cube, ABCD-A1B1C1D1, with the length of the edge as 1. How to draw this cube and establish a space rectangular coordinate system at the same time. Point A on the bottom is the origin of the coordinate system, the line AB is on the x-axis, the line AD is on the y-axis, and the line AA1 is on the z-axis.

After establishing the spatial rectangular coordinate system, the coordinates of each vertex of the cube are known.How can I get the coordinates of a vector composed of any two vertices? Can I get it automatically?

For example, if you input vector AB, you can get the coordinate AB of vector AB={1,0,0}and so on...

Answer 1

reg = Parallelepiped[{0, 0, 0}, {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}];
pts = MeshPrimitives[reg, 0];
g = HighlightMesh[
  reg, {Labeled[0, "Index"], Style[2, Directive[White, Opacity[.1]]]}]

• For two points,for example 6 and 4.

v[6, 4] = pts[[;; , 1]][[6]] - pts[[;; , 1]][[4]]
With[{i = 6, j = 4}, 
 Graphics3D[{{White, Lighting -> {{"Ambient", White}}, g}, Red, 
   Arrow[{pts[[;; , 1]][[i]], pts[[;; , 1]][[j]]}]}, Boxed -> False]]

• Labeled

HighlightMesh[reg, {Labeled[{0, 1}, Style["A", Red, 15]], 
  Labeled[{0, 2}, Style["B", 15]], Labeled[{0, 4}, Style["D", 15]], 
  Labeled[{0, 5}, Style["A1", 15]], Labeled[{0, 6}, Style["E", 15]], 
  Style[2, Directive[White, Opacity[.1]]]}]

• We can also use Graphics3D+Text to draw such picture, but here we let it be a exercise.