Course: Calculus III
Topic: Vector-Valued Functions
Subtopic: Normal and Binormal Vectors


Consider an object moving along a curve in space. Recall that two ways to change the velocity of an object (to accelerate) are to change its speed and change its direction. Curvature tells how fast the curve turns. At a point P on the curve, the normal vector describes the direction in which the curve turns. The binormal vector describes the twisting or torsion (denoted by tau, τ) of the curve.

The tangent vector T and the normal vector N may provide insight into how moving objects accelerate. Changing the speed produces acceleration in the direction of T and changing the direction produces acceleration in the direction of N. The binormal vector B is orthogonal to T and N. Investigating these three vectors and the three planes each pair spans as an object moves along a curve in space will provide much information about the direction, acceleration, and orientation of the object.

There is a lot going on here, so plan to spend quality time with the concepts making connections between the visuals of points moving and accelerating on curves, the vectors associated with them, and the physical applications these motions can help analyze.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: principal unit normal vector, tangential and normal components of the acceleration vector, unit binormal vector, torsion (tau, τ), TNB frame, mutually orthogonal vectors, osculating plane, normal plane, rectifying plane

Formulae to have in your notes: The following list is a screenshot from Briggs' Calculus: Early Transcendentals (3rd ed., ch. 14.6).

Supplemental Resources (optional)

Paul's OL Notes - Calc III: Tangent, Normal, and Binormal Vectors (study "Normal Vectors" and "Binormal Vectors" sections)

James Sousa's MathIsPower4U - Calc II: Determining the Binormal Vector