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 is found by vector addition.

Strategy: 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. (Figure below) Let r_o = < x_o, y_o, z_o > be a vector

from the origin to the point P_o (x_o, y_o, z_o) 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 r_o to r .

Then by vector addition: (Key step in determining the equation for the line)

r = r_o + tV ( This is the equation of the line in vector form. )

Equation of the line, L, in scalar form: x = x_o + at, y = y_o + bt, z = z_o + ct

The equation of the line, L, in “symmetric” form is obtained from the scalar form by

solving for t. The result is:

[x - x_o] / a = [y - y_o] / b = [z - z_o] / c = t

Click here for an example.

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