Lines in Space (Click here to skip to Planes in Space)
In a Nut Shell: There are three types of lines in space. Those that are parallel, those
that intersect, and those that are skew (neither parallel nor intersect). The equation of
a line in space can be found by vector addition as described below.
Let r = < x, y, z > be a vector from the origin, O, to an arbitrary point P(x, y, z)
on a line, L, in space. Let ro = < xo, yo, zo > be a vector from the origin to the
point Po (xo, yo, zo) on the same line, L. Let V be a vector < a, b, c > parallel
to the line L. Let t be a constant parameter.
Now let tV be a vector along (or parallel to line L) such that
tV = ta i + tb j + tc k is the vector from ro to r .
Then by vector addition: (Key step in determining the equation for the line)
r = ro + tV (this is the equation of the line in vector form)
Equation of the line, L, in scalar form: x = xo + at, y = yo + bt, z = zo + ct
The equation of the line, L, in “symmetric” form is obtained from the scalar form by
solving for t. The result is:
[x - xo] / a = [y - yo] / b = [z - zo] / c = t
Copyright © 2011 Richard C. Coddington
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