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Programming the Distance Formula

```
Date: 03/26/2001 at 22:26:50
From: Jean-Sebastien Payette

I have a problem similar to one of your archived explanations. I would
like to know how to use the concept you explained in "Using the
Distance Formula" at:

http://mathforum.org/dr.math/problems/calphool5.3.97.html

to be able to find the point on a line that's distance X from a given
point, but in a computer-based programming language.
```

```
Date: 03/26/2001 at 23:22:39
From: Doctor Peterson

Hello, Jean-Sebastien.

There is a considerably simpler way to do this than what was shown in
that answer. If you are at all familiar with vectors, you can follow
my reasoning; otherwise just take the result.

Suppose we know two points A (x1,y1) and B (x2,y2) that determine a
line, and want to find a point on that line that is d units from A, in
the direction of B. Then the vector AB = (x2-x1,y2-y1) points in the
right direction; its length is |AB| = sqrt((x2-x1)^2 + (y2-y1)^2). If
we divide vector AB by its length, we get a unit vector AB/|AB| in the
right direction; and if we then multiply this vector by length d, we
get the vector AC that takes us to the desired point.

This means that point C is:

C = A + AC = A + AB * d/|AB|

(x2-x1,y2-y1)*d
= (x1,y1) + ---------------------------
sqrt((x2-x1)^2 + (y2-y1)^2)

(x2-x1)d                        (y2-y1)d
= (x1 + -------------------------, y1 + -------------------------)
sqrt((x2-x1)^2+(y2-y1)^2)       sqrt((x2-x1)^2+(y2-y1)^2)

This should be easy to program.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculators, Computers
High School Coordinate Plane Geometry
High School Geometry

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