@acer Ok. Yes if you enter two lists or two vectors, yes your first answer accepts that. I hadn't thought of that. In general I would say that the two main sources for generating the  3D lines are,
1) manual generation which usually starts of as a pair of points and a procedure produces the line.
2) operations on 3D lines to produce other 3D lines. e.g the perpendicular bisectors of two intersecting 3d lines. 
I will have to dwell on this format choice.

@acer Yes that is what I am trying to do. Reasoning:- it is a list and vector for 3D lines. I have being using this format

Vector[column](3, [7 + alpha, 8 + 2*alpha, 9 + 3*alpha])

which is good but gets mixed up with other vectors at times. I was just going to enclose that in a set. I have done that with 2D line vectors  i.e a*x+b*y+c ->{<a,b,c>} as that stops data colisions in the procedures. The other problem with the current 3D line vector structure is that it is fiddly to seperate the list component from the vector component. I picked on using a set because I have so many things already structured as lists of things. 
I like that it can handle { l , v } or { v , l  }.. Order here is not an issue as they are different types. I do understand that you consider it suboptimal coding. 

  The satisifies is there because that is how sombody answered a question about 3 years ago on checking for a "3 list". It worked so I never changed it. Which I admit is a bad policy of using something without usderstanding it properly which has caught me out badly at times months doun the line.

@dharr Interesting approach. Vote up for that.

Edit:- What is the Permanent used for?

@acer Thank you. We can leave it at that for now. There are other places where I have being using radnormal but evala might be better . That will take some testing. 

@acer  Yes that is basically what I am after. I am interested in you modifications.
Basically I came up with that code about two years ago and I am sure I tried a lot of things to get it to work.
So I gather your use of normal can help with factoring/reducing each element of the matrix.
Is setting tgdc := numer(SV1[1]); the key component here?
I like _nresults Will have to see where else I can use that concept.
"pps. I didn't much alter your approach (lcm & gcd, say), but one might..."  If you have a better approach I am interested. AFAIR I used the frondend(lcm.... because often there are square roots.

@dharr Wow. That is great. Yes too many relationshiips to chose from. I reckon in the paper the substitutions are probably hand picked. I really like the 

@acer You are correct and should have seen that myself as you have pointed that out to me before. Even though this is being applied to my own use of the plotting routines inside a package this works perfectly. 
    The "square", sorry I should have looked up exact names.
 The colour vs color. That is just to handle the different spellings of the word. 

Can't say I had ever tried that before without a document open. But the same occurs in Windows 10 with Maple 2026.1. Also I can't add to the favourites without a document open. Same in 2024.2

@dharr  I know this behaviout was specifically mentioned with the release of 2025 and the now interface. One work around is to use brackets when  in document mode.

A := 1/(2! + 3 + 7/5);
                                 5 
                            A := --
                                 32

B^((2 + 8)/4);

@C_R Agreed. I had forgotten about using // in the new interface.

@C_R That explains my confusion. I had probably only ever used tables in document mode.

@dharr Thank you. Is this this a new feature of tables that they are text by defaul? I haven't used them much in the past.

@acer That is exactly what I was trying to achieve with the test proc. It is to differentiate between types of inputs. I have a rather large procedure for creating lines and planes with optional parallel or perpendicular lines through a point. But the procedure will not run outside of the package. I have been trying to make it more efficient.  Attached a pdf of the help page for it just to give you an idea of what it does. In the meantime this evening I figured out a variation of what you have done. Yours is far neater though.

l2::{satisfies(s->`ifelse`((type(s,`=`) and not(type(rhs(s),list))),true,false )  or type(s,`+`) or type(s,`*`) or type(s,symbol) or type(s,procedure))})   

Help_Lines_and_Plane.pdf

@acer  I had another typo from editing the module. It works now. The additional question didn't arise untill I considered how I would approach changing the basic approached I took in the RationalTrigonometry pagkage which has about 60 procedures in it. Thank you for you help.

@acer Sorry. I meant I renamed the TM to TestModule and then tried using TM:-Cartvars:=Cartvars inside it instead of TestModule:-Cartvars:=Cartvars  for example

The real package is called RationalTrigonometry.  inside that I want to use RT:-Cartvars:=Cartvars  instead of RationalTrigonometry:-Cartvars:=Cartvars  if possible.