David Kastrup <dak@gnu.org> wrote:
Frankly, that's academic. It reminds me of the excessive amount of
energy placed into keeping the mathematics(!) of location vectors and
difference vectors separate.
You can definitely represent positions and differences in positions as
vectors, but there is an actual difference, most notably what the sum
of two vectors is.
If you have two difference vectors (from P to Q, and from Q to R) and
add them together, you get the difference vector from P to R.
If you have two points P and Q, P+Q is a point in the plane, but which
point it designates is a bit more complicated. (When you take the four
points P, Q, P+Q, and the origin, you get the corners of a
parallelogram, which might be degenerate (i.e., have no area).) Of
course, you can subtract P from Q to get the difference vector, which
*does* have a meaningful interpretation.
You can also add different types; if you add the point P to the
difference vector from Q to P to it, you get the point Q.
When you ask the question, "What does the sum of two vectors mean?"
you have to know what they represent, and therein the difference
actually lies. The numerical computation is, of course, the same.