Hcfyk

documentclassstandalone
usepackagetikz
usetikzlibrarycalc,intersections
usepackageamsmath

begindocument

begintikzpicture[scale=0.5]
draw[<->] (0,12) node[above]$w$ |- (12,0) node[right]$q_L$;

draw[dotted, name path = ACL] (0,0) plot [domain=0.7:9] (x,0.75*x + 0.8) node[label=right:AC$_L$ = S$_L$](A1);
coordinate (H) at ($(5.6/0.75,5.6) - (0.8/0.75,0)$);
draw[thick, blue, name path = ACLt] let p1 = (H) in (0,y1) node[left, red]$w_u$ -- (x1,y1) --(A1);
endtikzpicture 

enddocument

My problem is as follows:

• I have attempted to define a coordinate (A1) at the end of the first line (name path ACL).


• I then use this coordinate (A1) as the last coordinate in the second line (name path ACLt).

It seems that the coordinate is not truly at the end of the line, since the dotted line is extending slightly beyond the thick blue line:

Why is this occurring?

Well, Zarko has turned my comment into "his" answer, but here it is again: use coordinate instead of node because the latter has a finite size (by default). And my answer and comment come with an arguably simpler construction of the path.

documentclassstandalone
usepackagetikz
usetikzlibrarycalc,intersections
usepackageamsmath

begindocument

begintikzpicture[scale=0.5]
draw[<->] (0,12) node[above]$w$ |- (12,0) node[right]$q_L$;

draw[dotted, name path = ACL] (0,0) plot [domain=0.7:9] (x,0.75*x + 0.8) 
coordinate[label=right:AC$_L$ = S$_L$] (A1);
coordinate (H) at ($(5.6/0.75,5.6) - (0.8/0.75,0)$);
draw[thick, blue, name path = ACLt] (0,0|-H) node[left, red]$w_u$ -- (H) --(A1);
endtikzpicture 

enddocument

calculation of coordinate is sufficient accurate, problem is that for it you use node, replace node with coordinate and result will become as you wish:

documentclassstandalone
usepackagetikz
usetikzlibrarycalc,intersections
usepackageamsmath

begindocument

begintikzpicture[scale=0.5]
draw[<->] (0,12) node[above]$w$ |- (12,0) node[right]$q_L$;

draw[dotted, name path = ACL]
 plot [domain=0.7:9] (x,0.75*x + 0.8) coordinate[label=right:AC$_L$ = S$_L$] (A1);
coordinate (H) at ($(5.6/0.75,5.6) - (0.8/0.75,0)$);
draw[thick, blue, name path = ACLt] let p1 = (H) in (0,y1) node[left, red]$w_u$ -- (x1,y1) --(A1);
endtikzpicture

enddocument

difference between node and coordinate arise since node has some width regardless if it is empty. it is determined by default value of inner sep. if you set it to zero: inner sep=0pt the result will be the same.

off-topic: in your diagram you not use intersections library, so you can remove line names from code (as i do in aboveo mwe)