Section 12.8

Tangent, Normal, and Binormal Vectors

The "Moving Trihedron" frame of reference that travels along a curve.

1

Introduction

At any point on a smooth 3D curve, we can define three mutually perpendicular unit vectors:
T (Direction of motion), N (Direction of turning), and B (Cross product). Together, they form the TNB Frame.

Interactive: The TNB Frame

T (Tangent) points forward. N (Normal) points inward (turning). B (Binormal) is orthogonal to both.

2

Key Formulas

Unit Tangent (T)

Direction of velocity.

Unit Normal (N)

Direction of turning.

Binormal (B)

Perpendicular to motion plane.

3

Worked Examples

Example 1: Finding T(t) (Level 1)

Find for .

1. Derivative r'(t)
2. Magnitude |r'(t)|

3. Divide

Example 2: Finding N(t) (Level 2)

Find for the same helix.

1. Derivative T'(t)

From Ex 1: .
.

2. Magnitude |T'(t)|

.

3. Divide
.

Example 3: Geometric Insight (Level 3)

Interpret for the helix.

We found .

  • Vector points from back toward the z-axis (center of the spiral).
  • The z-component is 0, meaning the "turning" happens purely horizontally.
  • Key Takeaway: always points toward the center of curvature.
4

Practice Quiz

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