INTRODUCTION TO VECTOR CALCULUS CLASSIFICATION OF VECTORS Broadly vectors can be classified into two categories– (i)

Axial Vectors : Where a vector has rotational motion lying along the normal to the plane of rotation of the body and remains unchanged under inversion. e.g.: Torque, angular momentum etc.

(ii)

Polar Vector : Where a vector has linear motion in a particular direction but changes under inversion or reflection. e.g. displacement, position vector, velocity etc.

Some special vectors (i)

Unit vector : It is a vector with unit magnitude and characterizes the direction of the vector mathematically it is denoted by

 A ˆ = A A In Cartesian coordinate system, let us choose three unit vectors along three mutually perpendicular axes as ˆi, ˆj and kˆ in x, y and z directions respectively. Then any  arbitrary vector A can be expressed as

 ˆ ˆ ˆ A = A x i  A y j  Az k  where Ax, Ay and Az are called the components of A in x, y and z directions.

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Introduction to Vector Calculus y

y

A

ˆj kˆ

x

ˆi

A z kˆ

x

0

A y ˆj

A x ˆi

z

z

 The magnitude of A is given using parallelogram law  A = A 2x  A 2y  A z2  hence unit vector along A is given by A x iˆ  A y ˆj  A z kˆ

ˆ = A

A 2x  A 2y  A z2

Direction cosines  The cosines of the angles, which A makes with x, y and z axis are called direction    cosines of the vector. If l, m and n are the direction cosines along ox, oy and oz axes, then

and

Ax cos    l= A

or

 Ax  l A

Ay  m = cos   A

or

 Ay  m A

Az cos    n= A

or

 Az  n A

2 A A 2x  A 2y  A 2z   2 1 2 l2  m2  n 2 = A A

Introduction to Vector Calculus

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2 2 2 l2  m2  n 2 = cos   cos   cos   1   ˆ ˆ ˆ A = A li  m j  n k

so



and



and the unit vector

 A   liˆ  m ˆj  n kˆ aˆ = A (ii)

Null Vector : Any vector with magnitude zero is called null vector. It is collinear  with every vector and denoted by O .

(iii)

Collinear or parallel vector : When vectors are parallel, then these are collinear vectors, whatsoever their magnitudes may be. Direction of there vectors may be some or opposite. A A

B A

B B

C C

D

C

When any scalar is multiplied to any vector then the resultant vector becomes     collinear with original one. e.g. B  A i.e. B vector is  times A with same  direction as of A . (iv)

Coplanar vectors : When vectors lies in the same geometrical plane they are called coplanar vectors. Otherwise these are called non-coplanar vectors.

(v)

Like vectors : The collinear vectors with same sense of direction irrespective of magnitude are called like vectors.

(vi)

Reciprocal vectors : When the magnitude of a vector is reciprocal to the magnitude of other vector with same direction then it is called reciprocal vector. It is written   aˆ 1 as  i.e. A 1   where aˆ is the unit vector along the direction of A . A A

Product of vectors (i)

Scalar product or dot product :

When the result of product of two vectors is a scalar quantity then this product is known as scalar (or dot) product of the given vectors.

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Introduction to Vector Calculus

Mathematically it is obtained by multiplying the magnitudes of the vectors with cosines of the angle between them   i.e. A.B = A B cos     A.B = A B cos  P

B 0

 B cos 

Q

A

y

Alternatively scalar product may be defined as multiplication of one vector with component of another in the direction of first. In case of Cartesian unit vectors ˆi.iˆ = i . i cos 0  1.1  1  ˆj. ˆj  kˆ . kˆ

and

ˆi. ˆj = i j cos 90  0  ˆj . kˆ  iˆ . kˆ

if

 ˆ ˆ ˆ A = Ax i  Ay j  Az k

 ˆ ˆ ˆ B = Bx i  B y j  B z k   then A.B = A x B x  A y B y  A z B z     (a) Scalar product obeys commutative law i.e. A.B  B.A        (b) It obeys distributive law i.e. A. B  C  A.B  A.C and





Two non-zero vectors are orthogonal or perpendicular when  = 90° i.e. cos  = 0   then A.B = 0 (c)

similarly two vectors are collinear when  = 0 or  i.e. cos  = ± 1   then for  = 0, A.B = AB   and for  = , A.B = –AB Physical examples– (i)

  Work done W  F. ds

Introduction to Vector Calculus

(ii) (iii)

(iv)

(ii)

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  Power = F.v    Magnetic flux of a magnetic field = B . ds where B is magnetic flux density over  an area ds .    Electric flux of an electric field = E. ds where E the electric field intensity  through elementary area ds .

Vector product or cross product

When the product of two vectors is a vector quantity, then the product is called vector product or cross product mathematically it is written as    C = A B   = A B sin . nˆ where 0    

  here nˆ is the unit vector in the direction of normal to the plane containing A and B such      that A, B and C from a right handed coordinate system with rotation from A to B . for ˆi, ˆj and kˆ . ˆi  ˆi = i i sin 0  0  ˆj  ˆj  kˆ  kˆ

and

ˆi  ˆj = i j sin 90  1  ˆj  kˆ  kˆ  ˆi

but

ˆj  ˆi = j i sin  90    1  kˆ  ˆj  ˆi  kˆ

if and then

 ˆ ˆ ˆ A = Ax i  Ay j  Az k  ˆ ˆ ˆ B = Bx i  B y j  B z k     A  B = A B sin  nˆ

=

or

ˆi Ax

ˆj Ay

kˆ Az

Bx

By

Bz

  A  B = A y Bz  A z B y iˆ   A x Bz  A z Bx  ˆj  A x B y  A y B x









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Introduction to Vector Calculus

  AB   nˆ = A  B

and

  AB sin  =   A B

hence

where

 A =

A 2x  A 2y  A z2

and

 B =

B2x  B 2y  B z2

     if the rotation from A to B is anti clockwise then C  A  B is +ve. and if rotation is    clockwise then C  A  B is –ve. A

C  B C



A

0

Properties (a)

Cross product is not commutative         A  B   B  A but A  B   B  A

(b)

       It is distributive i.e. A  B  C  A  B  A  C

(c)

If two vectors are collinear or parallel then





 = 0 or  then and (d)

sin  = 0   AB = 0

Two vectors are perpendicular then  = 90°     so A  B = A B nˆ

B

Introduction to Vector Calculus

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Some examples (i)

Moment of forces  =   rF 

(iii)

     Angular momentum L  r  p  m  r  v     Linear velocity v    r

(iv)

force on a charged particle

(ii)

   F = q v  B where q is in coulombs.



(v)



Force on a charged particle moving through electric and magnetic field is     F = q E  v  B . this is known as Lorentz force.





Scalar Triple product When a vector is scalarly multiplied with the cross product of other two vectors then the result is called scalar triple product.              A B C  = A . B  C  B. C  A  C . A  B  



   A. BC





=





Ax

Ay

Az

Bx

By

Bz

Cx

Cy

Cz







Vector triple product When any vector is vectorily multiplied with vector product of other two vectors taken in cyclic order then the result is known as vector triple product.          A  B  C = B A .C  C A . B

     B  C  A     C   A  B

          = C  B.A   A  B.C        = A  C.B   B  C.A 

Properties (i) (ii)

         A  B  C  B C  A  C  A  B  0











              A  B  C   C   A  B   A  B.C   B  A.C 

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Introduction to Vector Calculus









 

 

 



 

 

(iii)

 A  B .  C  D    A .C   B.D    A .D   B.C 

(iv)

    A  B C  D

 



 





   B  A.  C  D     A.B  C  D 

Vector differentiation This is the limiting value of ratio of a vector to the change of a scalar as the change tends to zero is called vector differentiation.

f (u + u)

f = f (u + u) – f (u)

f (u)

 f df = lim  u  0  u du     u  u   f  u  = lim u  0 u

Properties (a)

(b)

(c)

(d)

(e)

(f)

  d   dA dB A B   du du du   d   dA   dB A .B  . B  A. du du du    dB dA  d   A B  A   B du du du       dC    dB   dA   d    A. B  C  A.  B   C  . BC   A.  du du   du  du        dC    dB   dA     d  A  B C  A   B  C   BC  A  du du   du  du   d  da ds a s  dt ds dt













 

 













Introduction to Vector Calculus

(g)

   d    de de   de  e.e   e.   e  2e  dt dt dt dt  d 2  de e  2e. or dt dt  d  da  a  t     dt dt

 

(h)

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