\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \)
17calculus
Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Gradients
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
17calculus on YouTube
More Math Help
Tutoring
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > vector functions > principal unit normal vector

Vector Functions - Principal Unit Normal Vector

on this page: ► conceptual understanding     ► how to calculate     ► shortcut in 2-dim

On this page, we explain how to find a normal vector to a curve. In general, there are many normal vectors. However, one of them, the principal unit normal vector, is the one we focus on since this normal vector tells in what direction the curve is 'curving'. When we get to curvature, we will use this vector to also determine how 'tight' the curve is.

Conceptual Understanding

Plot 1

Before we get into the details of how to calculate the principal unit normal vector, let's look at an example to get a conceptual understanding of what we are talking about.
Plot 1 shows a curve (in black), the unit tangent vector (in green) and a normal vector (in blue) at the point \((1,2)\). Let's discuss each curve individually.

Basic Curve - - The basic curve (in black) is traced out starting at \((0,0)\) and going up and to the right, as shown by the arrow on the curve.
Unit Tangent Vector - - The unit tangent vector (in green) is tangent to the curve at the point of interest. Of course, as you know from basic calculus, there are two vectors tangent to the curve at each point but we are only interested in the unit tangent vector pointing in the direction of motion.
Principal Unit Normal Vector - - A normal vector (in blue) is shown in Plot 1. There are actually two normal vectors, the one we show and another in the opposite direction (not shown). We are interested only in the principal unit normal vector, which is the normal vector of length one that points to the inside of the curve.

How To Calculate

To calculate the principal unit normal vector, we use the unit tangent vector \(\vec{T}(t)\). The equation is \(\displaystyle{ \vec{N}(t) = \frac{d\vec{T}/dt}{ \| d\vec{T}/dt \| } }\)
We normally use \(\vec{N}\) to specify this vector within the context of vector functions. Here is quick video proving that \(\vec{T}\) and \(\vec{N}\) are orthogonal.

MIP4U - Proving the Unit Normal Vector Formula [3min-41secs]

Shortcut in 2-dim

Remember from vector algebra that if you have a vector, you can get a vector normal to it by switching the x and y components and changing the sign of one of them. For example, for the vector \(\vec{v} = \langle a,b \rangle\), vectors \(\vec{w} = \langle -b,a \rangle\) and \(\vec{z} = \langle b,-a \rangle\) are both normal to \(\vec{v}\). (To check this, take the dot products and make sure you get zero.) However, there are two vectors and it is not possible to tell which is the principal normal vector. But, if you are allowed to, you can usually graph the original function and the vectors and pull the information off the graph.
Notes - -
1. This works only in 2-dim, i.e. for space curves (3-dim) you must use the vector equation given above for \(\vec{N}\).
2. As with everything on this site, make sure to check with your instructor to see if they allow you to use this shortcut.

Here is a quick video clip discussing the principal unit normal vector and this shortcut.

MIP4U - Determining the Unit Normal Vector [2min-41secs]

Okay, time for some practice problems. After that, your next logical topic is how to describe the acceleration vector in terms of the unit tangent and principal unit normal vector.

next: acceleration vector →

Search 17Calculus

Practice Problems

Instructions - - Unless otherwise instructed, find the principal unit normal vector of the curve. If a point is given, also find the principal unit normal vector at that point.

Level A - Basic

Practice A01

\(\vec{r}(t)=\langle \cos(t),\sin(t),t \rangle\), \(t=\pi/4\)

answer

solution

Practice A02

\(\vec{r}(t)=\langle t^3,2t^2 \rangle\), \(t=1\)

answer

solution

Practice A03

Calculate \(\vec{T}(0)\) and \(\vec{N}(0)\) for \(\vec{r}(t)=\langle t\sqrt{2},e^t,e^{-t}\rangle\).

answer

solution

Practice A04

\(\vec{r}(t)=\langle t,3\cos(t),3\sin(t)\rangle\)

answer

solution

Real Time Web Analytics
menu top search practice problems
17
menu top search practice problems 17