Section 12.2

Equations of Lines

Defining lines in space using a starting point and a direction vector.

Introduction

In 2D, we used slope . In 3D, "slope" is a vector. To define a line, we need a point and a parallel direction vector . The equation works by starting at and stretching by a parameter .

Interactive: Line Vector Construction

The green line is traced by . Drag to move along the line!

Key Formulas

Vector Equation

Through parallel to :

Parametric Equations

Breaking it down by component:

Symmetric Equations

Solving for t (if ):

Worked Examples

Example 1: Finding Equations (Level 1)

Find the equation of the line through parallel to .

Vector Form

Parametric Form

Symmetric Form

Since , we separate the part:

Example 2: Line Through Two Points (Level 2)

Find the line connecting and .

Hint: First find the direction vector .

1. Direction Vector

2. Choose P as base

3. Parametric Equations

Example 3: Intersection with Plane (Level 3)

Where does the line intersect the -plane?

Concept: The -plane is defined by .
Set y component to 0:
Find Coordinates (t=4):

(Check!)

Intersection point: