Tikz intersection without extending the paths till they intersect

Yes, you can, at least when the intersecting curves are lines. The idea is to use intersection in a let ... in action. I saw this construction in AndréC answer Drawing a circle through 3 non-collinear points.

I consider that two lines are given through a point and a vector for each---A and v for the first and B and w for the second. We need their meeting point C.

documentclass[tikz, border=1cm]{standalone}
usetikzlibrary{calc, intersections}

begin{document}

begin{tikzpicture}[every node/.style={scale=.8}]
  % points and directions
  path
  (0, 0) coordinate (A) (80:1) coordinate (v)
  (2, 0) coordinate (B) (100:1) coordinate (w);

  % intersection point
  path let
  p1 = (A),
  p2 = ($(A)+(v)$),
  p3 = (B),
  p4 = ($(B)+(w)$),
  p5 = (intersection of p1--p2 and p3--p4)
  in
  (p5) coordinate (C);
  
  draw (A) -- (B) -- (C) -- cycle;

  draw[blue, very thick, ->] (A) -- +(v) node[above left] {$v$};
  draw[blue, very thick, ->] (B) -- +(w) node[above right] {$w$};
  
  foreach P/pos in {A/below left, B/below right, C/above left}{%
    draw[fill=white] (P) circle (1.2pt) node[pos] {$P$};
  }
end{tikzpicture}
end{document}

There is a syntax that still works for the intersection of two lines. It is no longer documented since version 3 of TikZ, but you will find its syntax in the old manuals version 1 and 2.

Its interest is that the intersection is found even if the segments are not intersecting, it is enough that the lines carrying them are not parallel.

It does not need any TikZ library to be used.

documentclass[tikz, border=1cm]{standalone}

begin{document}%

begin{tikzpicture}
coordinate (A) at (0,0);
coordinate (B) at (2,0);    
path(A)--++(80:6cm)coordinate(u);
path(B)--++(100:6cm)coordinate(v);
%path[name intersections={of=AC and BC, by=C}];
draw (A)--(B)--(intersection of A--u and B--v)--cycle;
end{tikzpicture}

end{document}