Lines in Space

The central idea

A line in $\mathbb R^3$ through point $\mathbf r_0$ with nonzero direction $\mathbf v$ is $\mathbf r(t)=\mathbf r_0+t\mathbf v$. Component form is $x=x_0+at$, $y=y_0+bt$, $z=z_0+ct$ for $\mathbf v=\langle a,b,c\rangle$. Symmetric form divides by nonzero direction components only.

Definitions, hypotheses, and notation

A line segment from $A$ to $B$ uses $\mathbf r(t)=A+t(B-A)$ with $0\le t\le1$; the unrestricted parameter gives the entire line. Distance from a point $P$ to a line can be obtained by removing from $P-A$ its projection onto the direction, or in three dimensions by $\|(P-A)\times\mathbf v\|/\|\mathbf v\|$.

To test whether two lines intersect, solve $A+t\mathbf v=B+s\mathbf w$ for two parameters. An inconsistent three-equation system with nonparallel directions identifies skew lines. Parallel directions still require a point-membership test to distinguish coincident from distinct lines.

Conceptual meaning

The parameter moves from the base point along all scalar multiples of one direction. Different base points on the same line and nonzero scalar multiples of the direction produce different equations for the same geometric set.

A dependable method and decision rule

1. Identify one point and a nonzero direction vector.
2. Write vector or component equations with one shared parameter.
3. For two points, subtract in a consistent order to get a direction.
4. Solve the same parameter across coordinates when testing membership or intersections.
5. Compare directions and consistency to classify parallel, intersecting, coincident, or skew lines.

Fully worked example

Common mistakes and why they fail