Linear geometry with vectors

Here I'll summarize the formulas in Section 11.5 of the textbook that can be made simpler by doing arithmetic with points and vectors (instead of just with vectors as the book does) or by using the two-dimensional cross product (instead of only the three-dimensional cross product as the book does).

A parametric equation for the line through a point P0 in the direction of a nonzero vector v is

P = P0 + tv,
where t is the parameter and P = (x, y) or P = (x, y, z) is a point on the line.

Similarly, a parametric equation for the line through points P1 and P2 is

P = P1 + t(P2 − P1).

A nonparametric equation for the line through P0 in the direction of v in 2 dimensions is

(P − P0) × v = 0.

Similarly, a system of equations for the line through P0 in the direction of v in 3 dimensions is

(P − P0) × v = 0.
(The only difference is whether the zero on the right-hand side is the scalar 0 or the vector 0.)

The distance from a point S to the line through P0 in the direction of v is

|(S − P0) × v̂| = |(S − P0) × v|/|v|.

Similarly, the distance from S to the line through P1 and P2 is

|(S − P1) × (P2 − P1)|/|P2 − P1|.

An equation for the line (in 2 dimensions) or plane (in 3 dimensions) through P0 and perpendicular to a vector n is

(P − P0) ⋅ n = 0.

The distance from S to the line or plane through P0 and perpendicular to n is

|(S − P0) ⋅ n̂| = |(S − P0) ⋅ 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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