A parametric equation for the line
through a point *P*_{0}
in the direction of a nonzero vector **v** is

whereP=P_{0}+tv,

Similarly, a parametric equation for
the line through points *P*_{1} and *P*_{2} is

P=P_{1}+t(P_{2}−P_{1}).

A nonparametric equation for
the line through *P*_{0} in the direction of **v**
in 2 dimensions is

(P−P_{0}) ×v= 0.

Similarly, a system of equations for
the line through *P*_{0} in the direction of **v**
in 3 dimensions is

((The only difference is whether the zero on the right-hand side is the scalar 0 or the vectorP−P_{0}) ×v=0.

The distance from a point *S*
to the line through *P*_{0} in the direction of **v** is

|(S−P_{0}) ×v̂| = |(S−P_{0}) ×v|/|v|.

Similarly, the distance from *S*
to the line through *P*_{1} and *P*_{2} is

|(S−P_{1}) × (P_{2}−P_{1})|/|P_{2}−P_{1}|.

An equation for
the line (in 2 dimensions) or plane (in 3 dimensions)
through *P*_{0} and perpendicular to a vector **n** is

(P−P_{0}) ⋅n= 0.

The distance from *S*
to the line or plane
through *P*_{0} and perpendicular to **n** is

|(S−P_{0}) ⋅n̂| = |(S−P_{0}) ⋅n|/|n|.

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This web page was written in 2016 by Toby Bartels, last edited on 2016 March 11. Toby reserves no legal rights to it.

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`http://tobybartels.name/MATH-1700/2016WN/vecgeo/`

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