Section 12.6

Vector Functions

Describing motion and curves in 3D space using vector-valued functions .

Introduction

A vector function maps a scalar to a vector in space. As varies, the tip of the vector traces out a 3D curve. Common applications include describing the path of a particle moving through space.

Interactive: Tracing a Helix

. Drag to trace the path!

Key Formulas

Definition

Line Segment

From to :

for

Common Curves

Line: Linear functions of .
Circle/Ellipse: Involves in two components.
Helix: Circle in two components, linear in the third (e.g., rising spiral).

Worked Examples

Example 1: Finding Domain (Level 1)

Find the domain of .

Step 1: Check components separately

Step 2: Intersect

Combined: AND .
Interval notation: .

Example 2: Parameterizing a Line Segment (Level 2)

Find the vector equation for the segment from to .

Formula: for .

Combine components:
x:
y:
z:

Result:

Example 3: Identifying Curves (Level 3)

Describe the curve .

Look at x and y:
.
This means the curve lies on a Cylinder of radius 3 centered on the z-axis.
Look at z:
. The height increases linearly with angle .
Conclusion:
Points rotate around the cylinder while rising. This is a Circular Helix.