OneStopGate.Com
OnestopGate   OnestopGate


   Sunday, February 05, 2012 Login  

OnestopGate
Home | Overview | Syllabus | Tutorials | FAQs | Downloads | Recommended Websites | Advertise | Payments | Contact Us | Forum
OneStopGate

GATE Resources
Gate Articles
Gate Books
Gate Colleges 
Gate Downloads 
Gate Faqs
Gate Jobs
Gate News 
Gate SamplePapers
Training Institutes
GATE Overview
Overview
GATE Eligibility
Structure Of GATE
GATE Coaching Centers
Colleges Providing M.Tech/M.E.
GATE Score
GATE Results
PG with Scholarships
Article On GATE
GATE Forum

GATE 2010 Exclusive
Organizing Institute
Important Dates
How to Apply
Discipline Codes

GATE 2010 Syllabus
Aerospace Engg..
Agricultural Engg..
Architecture and Planning
Chemical Engg..
Chemistry
Civil Engg..
Computer Science / IT
Electronics & Communication Engg..
Electrical Engg..
Engineering Sciences
Geology and Geophysics
Instrumentation Engineering
Life Sciences
Mathematics
Mechanical Engg..
Metallurgical Engg..
Mining Engg..
Physics
Production & Industrial Engg..
Pharmaceutical Sciences
Textile Engineering and Fibre Science

GATE Study Material
Aerospace Engg..
Agricultural Engg..
Chemical Engg..
Chemistry
Civil Engg..
Computer Science / IT
Electronics & Communication Engg..
Electrical Engg..
Engineering Sciences
Instrumentation Engg..
Life Sciences
Mathematics
Mechanical Engg..
Physics
Pharmaceutical Sciences
Textile Engineering  and Fibre Science

GATE Preparation
GATE Pattern
GATE Tips N Tricks
Compare Evaluation
Sample Papers 
Gate Downloads 
Experts View

CEED 2010
CEED Exams
Eligibility
Application Forms
Important Dates
Contact Address
Examination Centres
CEED Sample Papers

Discuss GATE
GATE Forum
Exam Cities
Contact Details
Bank Details

Miscellaneous
Advertisment
Contact Us






My site is worth $44,432.
How much is yours worth?

Home » GATE Study Material » Engineering sciences » Elementary Vector Analysis

Elementary Vector Analysis

Looking for GATE Preparation Material? Join & Get here now!
Elementary Vector Analysis

Elementary Vector Analysis



In order to measure many physical quantities, such as force or velocity, we need to determine both a magnitude and a direction. Such quantities are conveniently represented as vectors.




The direction of a vector v in 3-space is specified by its components in the x, y, and z directions, respectively:

(x,y,z) or xi + yj + zk,

i = (1,0,0)
j = (0,1,0)
k = (0,0,1)

where i, j, and k are the coordinate vectors along the x-, y-, and z-axes.

The magnitude of a vector v = (x,y,z), also called its length or norm, is given by

|| v || = Ö
x2+y2+z2
 
 

 

Notes

 

  • Vectors can be defined in any number of dimensions, though we focus here only on 3-space.
  • drawing a vector in 3-space, where you position the vector is unimportant; the vector's essential properties are just its magnitude and its direction. Two vectors are equal if and only if corresponding components are equal.

  • A vector of norm 1 is called a unit vector. The coordinate vectors are examples of unit vectors.

     

  • The zero vector, 0 = (0,0,0), is the only vector with magnitude 0.

     

Basic Operations on Vectors


To add or subtract vectors u = (u1,u2,u3) and v = (v1,v2,v3),

add or subtract the corresponding coordinates:

 

u + v = (u1+v1,u2+v2,u3+v3)

u - v = (u1-v1,u2-v2,u3-v3)


To mulitply vector u by a scalar k, multiply each coordinate of u by k:

 

k u = (ku1,ku2,ku3)

 

Example

The vector v = (2,1,-2) = 2i + j - 2k has magnitude

|| v || =     ___________
Ö22 +12 -(-2)2
 
= 3.

Thus, the vector 1/3v = (2/3,1/3,-2/3) is a unit vector in the same direction as v.

In general, for v ¹ 0, we can scale (or normalize) v to the unit vector v/ ||v|| pointing in the same direction as v.

 

Dot Product

Let u = (u1,u2,u3) and v = (v1,v2,v3). The dot product u · v (also called the scalar procuct or Euclidean inner product) of u and v is defined in two distinct (though equivalent) ways:

 

u · v= u1v1+u2v2+u3v3 u · v is a number ì
í
î ||
u || || v || cosq if u ¹ 0, v ¹ 0 0 if u = 0 or v = 0 where 0 £ q £ p is the angle between u and v

 

Properties of the Dot Product

 

  • u · v = v · u

     

  • u · (v + w) = (u · v) + (u · w)

     

  • u · u = || u ||2

     

See if you can verify each of these!

 

Example


If u = (1,-2,2) and v = (-4,0,2), then

u · v
=
(1)(-4)+(-2)(0)+(2)(2)
 
=
-4+0+4
 
=
0
 

Using the second definition of the dot product with || u || = 3 and || v || = 2Ö5,

  u · v = 0 = 6Ö5cosq
so cosq = 0, yielding q = p/2.

Though we might not have guessed it, u and v are perpendicular to each other!

In general,

 

Two non-zero vectors u and v are perpendicular (or orthogonal) if and only if u ·v = 0.

 

Projection of a Vector


It is often useful to resolve a vector v into the sum of vector components parallel and perpendicular to a vector u.

Consider first the parallel component, which is called the projection of v onto u. This projection should be in the direction of u and should have magnitude || v||cosq, where 0 £ q £ p is the angle between u and v. Let's normalize u to u/|| u || and then scale this by the magnitude || v ||cosq:

 

Projection of v onto u
= (|| v || cosq) u
||u||
 
= ||v|| ||u|| cosq
||u||2
u
= v · u
||u||2
u
 

The perpendicular vector component of v is then just the difference between v and the projection of v onto u.

In summary,

 

projection of v onto u:
= v · u
||u||2
u
vector component of v
perpendicular to u:
= v - v · u
||u||2
u
 

 

Cross Product

Let u = (u1,u2,u3) and v = (v1,v2,v3). The cross product u × v yields a vector perpendicular to both u and v with direction determined by the right-hand rule. Specifically,

 

u × v is a vector
u × v = (u2v3-u3v2)i - (u1v3-u3v1)j + (u1v2-u2v1)k

It can also be shown that

 

|| u × v || = || u || || v || sinq for u ¹ 0, for v ¹ 0

where 0 £ q £ p is the angle between u and v.



Thus, the magnitude || u ×v || gives the area of the parallelogram formed by u and v.

As implied by the geometric interpretation,

 Non zero vectors u and v are parallel if and only if u × v = 0.

 

Properties of the Cross Product

 

  • u × v = - ( v × u)

     

  • u × ( v + w ) = (u × v ) + ( u × w )

     

  • u × u = 0

     

Again, see if you can verify each of these.

In the following Exploration, select values for the components of u and v. You will see u · v and u × v computed and u, v, and u × v displayed on a coordinate system.

 

Exploration

 

Key Concepts

Let u = (u1,u2,u3) and v = (v1,v2,v3).

 

  • Basic Operations, Norm of a vector

     

     
      u + v  
    =
    (u1+v1,u2+v2,u3+v3)
      u - v  
    =
    (u1-v1,u2-v2,u3-v3)
    k u  
    =
    (ku1,ku2,ku3)
    || v ||
    =
      Ö
    x2+y2+z2
     
     
     

     

  • Dot Product

    u · v

    nowrap="nowrap">u1v1+u2v2+u3v3 = ì
    í
    î
    || u|| ||vvcosqif u ¹ 0, v ¹ 0 0u = 0 or v = 0

    where 0 £ q £ p is the angle between u and v

    For u ¹ 0, v ¹ 0, u · v = 0 if and only if u is orthogonal to v.

     

  • Projection of a Vector

     

    projection of v onto u:
    = v · u
    ||u||2
    u
    vector component of v
    perpendicular to u:
    = v - v · u
    ||u||2
    u
     

  • Cross Product

     

    u × v = (u2v3-u3v2)i - (u1v3-u3v1)j + (u1v2-u2v1)k

    || u × v || = || u || || v || sinq for u ¹ 0, for v ¹ 0

    where 0 £ q £ p is the angle between u and v.

    For u ¹ 0, v ¹ 0, u × v = 0 if and only if u is parallel to v.

Discussion Center

Discuss/
Query

Papers/
Syllabus

Feedback/
Suggestion

Yahoo
Groups

Sirfdosti
Groups

Contact
Us

MEMBERS LOGIN
  
EmailId:
Password:

  Forgot Password?
 New User? Register!

INTERVIEW EBOOK
Get 9,000+ Interview Questions & Answers in an eBook. Interview Question & Answer Guide
  • 9,000+ Interview Questions
  • All Questions Answered
  • 5 FREE Bonuses
  • Free Upgrades
Exciting Offers
GATE RESOURCES
 
  • Gate Books
  • Training Institutes
  • Gate FAQs
  • GATE Exam, Gate 2009, Gate Papers, Gate Preparation & Related Pages


    GATE Overview | GATE Eligibility | Structure Of GATE | GATE Training Institutes | Colleges Providing M.Tech/M.E. | GATE Score | GATE Results | PG with Scholarships | Article On GATE | GATE Forum | GATE 2009 Exclusive | GATE 2009 Syllabus | GATE Organizing Institute | Important Dates for GATE Exam | How to Apply for GATE | Discipline / Branch Codes | GATE Syllabus for Aerospace Engineering | GATE Syllabus for Agricultural Engineering | GATE Syllabus for Architecture and Planning | GATE Syllabus for Chemical Engineering | GATE Syllabus for Chemistry | GATE Syllabus for Civil Engineering | GATE Syllabus for Computer Science / IT | GATE Syllabus for Electronics and Communication Engineering | GATE Syllabus for Engineering Sciences | GATE Syllabus for Geology and Geophysics | GATE Syllabus for Instrumentation Engineering | GATE Syllabus for Life Sciences | GATE Syllabus for Mathematics | GATE Syllabus for Mechanical Engineering | GATE Syllabus for Metallurgical Engineering | GATE Syllabus for Mining Engineering | GATE Syllabus for Physics | GATE Syllabus for Production and Industrial Engineering | GATE Syllabus for Pharmaceutical Sciences | GATE Syllabus for Textile Engineering and Fibre Science | GATE Preparation | GATE Pattern | GATE Tips & Tricks | GATE Compare Evaluation | GATE Sample Papers | GATE Downloads | Experts View on GATE | CEED 2009 | CEED 2009 Exam | Eligibility for CEED Exam | Application forms of CEED Exam | Important Dates of CEED Exam | Contact Address for CEED Exam | CEED Examination Centres | CEED Sample Papers | Discuss GATE | GATE Forum of OneStopGATE.com | GATE Exam Cities | Contact Details for GATE | Bank Details for GATE | GATE Miscellaneous Info | GATE FAQs | Advertisement on GATE | Contact Us on OneStopGATE |
    Copyright © 2012. One Stop Gate.com. All rights reserved Testimonials |Link To Us |Sitemap |Privacy Policy | Terms and Conditions|About Us
    Our Portals : Academic Tutorials | Best eBooksworld | Beyond Stats | City Details | Interview Questions | India Job Forum | Excellent Mobiles | Free Bangalore | Give Me The Code | Gog Logo | Free Classifieds | Jobs Assist | Interview Questions | One Stop FAQs | One Stop GATE | One Stop GRE | One Stop IAS | One Stop MBA | One Stop SAP | One Stop Testing | Web Hosting | Quick Site Kit | Sirf Dosti | Source Codes World | Tasty Food | Tech Archive | Software Testing Interview Questions | Free Online Exams | The Galz | Top Masala | Vyom | Vyom eBooks | Vyom International | Vyom Links | Vyoms | Vyom World
    C Interview Questions | C++ Interview Questions | Send Free SMS | Placement Papers | SMS Jokes | Cool Forwards | Romantic Shayari