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

Graphical or geometric meaning

The direction arrow can slide anywhere along the line without changing orientation. In three dimensions, two nonparallel lines need not meet: they can lie in different planes and be skew.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute $t=0$ and recover the stated base point.
• Differentiate $\mathbf r(t)$ and recover the direction.
• Use all coordinate equations when checking an intersection.

Parallel, intersecting, and skew are distinct

Write a line as $\mathbf r=\mathbf r_0+t\mathbf v$, where $\mathbf v\ne\mathbf0$. Two lines are parallel when their direction vectors are scalar multiples; they coincide only if a point from one also lies on the other. Nonparallel lines in three dimensions need not intersect: solving all three coordinate equations with two parameters distinguishes an intersection from skew lines. Using only two coordinates can produce parameter values that fail the third equation. The distance from a point to a line can be checked through the perpendicular component or a cross-product formula. When converting between vector, parametric, and symmetric forms, keep zero direction components separate because division by zero is not permitted. A direction vector has no fixed starting point, while $\mathbf r_0$ does.

Practice

1. Give a line through $(0,1,2)$ parallel to $\langle1,-1,3\rangle$.
2. When are two direction vectors parallel?
3. What does a zero direction component mean?

Connections and next steps

• Lines in Three Dimensions
• Planes
• Intersections of Lines and Planes