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The Vector Equation of a Line

Given points A and B in R3 (3D) with position vectors a and b, the vector from A to B is always b - a. We can use this vector to find the equation of the line through A and B. In case you've never come across a parametric equation before, I'll try and relate it to real life...
Suppose we are in a starship located at point A (relative to some fixed galactic cartesian coordinates) and we are required to set course for the space station located at point B. To do this we need to tell the ship the position vector it should have at any given time, t. We set t=0 when we have position vector a, and we want to arrive at the space station when t=1.
Let r(t) be our position vector at time, t, then for 0 O t O 1 our course starts at A and is along the vector from A to B = b - a. Hence the equation of our course is:
r(t) = a + t (b - a )
for 0 O t O 1

You can easily check that,
r(0) = a
r(1) = b
and, for instance, we reach the point halway between A and B when t = 1/2, so the halfway point is,
r(0.5) = (1/2) (a + b)
By letting t be any real number we can get to any point along the line through A and B.


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