Lines in Space

Definition

A line in three-dimensional space is the set of all points that can be reached by starting at a point $P_0$ and moving any distance along a fixed direction vector $\mathbf{v}$.

Why It Matters

Precise 3D line modeling is the difference between a successful rendezvous in orbit and a collision; it provides the coordinate-free framework needed to navigate and build in the complex three-dimensional reality of our universe.

Core Concepts

• Vector Equation: $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$, where $\mathbf{r}_0$ is the position vector of the fixed point and $t$ is a scalar parameter.

  • How to read: “The vector function r of t equals the vector r zero plus t times the vector v.”

  • Meaning: Start at $\mathbf{r}_0$, move along direction $\mathbf{v}$ scaled by parameter $t$—all points on the line.

  • Parametric Equations: $x = x_0 + tv_1, y = y_0 + tv_2, z = z_0 + tv_3$. These describe the coordinates individually as functions of the parameter $t$.

  • How to read: “The coordinate x equals x zero plus t v one, with similar linear forms for y and z.”

  • Meaning: Component form of the vector equation—each coordinate changes linearly with $t$.

• Distance from Point to Line: The distance $d$ from a point $S$ to a line passing through $P$ with direction $\mathbf{v}$ is given by: $d = \frac{|\vec{PS} \times \mathbf{v}|}{|\mathbf{v}|}$

  • How to read: “The distance d equals the magnitude of the cross product of vector P S and vector v, all divided by the magnitude of vector v.”
  • Meaning / when to use: Cross product gives parallelogram area; dividing by $|\mathbf{v}|$ yields perpendicular distance from point to line.

Connected Concepts

• Planes in Space
• Distance Between Points in Space
• Cross Product
• Vector-Valued Functions