Scalar and Vector Fields
• A scalar field is a function that gives us a
single value of some variable for every point
in space.
• Examples: voltage, current, energy, temperature
• A vector is a quantity which has both a
magnitude and a direction in space.
• Examples: velocity, momentum, acceleration and
force
Example of a Scalar Field
Scalar Fields
e.g. Temperature: Every location has associated
value (number with units)
3
Scalar Fields - Contours
• Colors represent surface temperature
• Contour lines show constant temperatures
4
Fields are 3D
•T = T(x,y,z)
•Hard to visualize
 Work in 2D
5
Vector Fields
Vector (magnitude, direction) at every point in
space
Example: Velocity vector field - jet stream
6
Vector Fields Explained
Examples of Vector Fields
Examples of Vector Fields
Examples of Vector Fields
Scalar and vector quantities
• Scalar quantity is defined as a quantity or
parameter that has magnitude only.
• It independent of direction.
• Examples: time, temperature, volume, density,
mass and energy.
• Vector quantity is defined as a quantity or
parameter that has both magnitude and
direction.
• Examples: velocity, electric fields and magnetic
fields.
• A vector is represent how the vector is oriented
relative to some reference axis.
• Graphically, a vector is represented by an arrow,
defining the direction, and the length of the arrow
defines the vector’s magnitude as shown in figure.
• Vectors will be indicated by italic type with arrow on
the character such as Ᾱ.
• Scalars normally are printed in italic type such as A.
• A unit vector has a magnitude of unity (â=1).
• The unit vector in the direction of vector Ᾱ is
determined by dividing A.

A
aˆ 
A
• By use of the unit vectors x̂, ŷ, ẑ along x, y and
z axis of a Cartesian system, a vector quantity
can be written as:

A  Ax xˆ  Ay yˆ  Az zˆ
• The magnitude is defined by

2
2
2
A  A  Ax  Ay  Az
• The unit vector is defined by

A Ax xˆ  Ay yˆ  Az zˆ
aˆ  
2
2
2
A
Ax  Ay  Az
Example 1
A vector Ᾱ is given as 2x̂ + 3ŷ sketch Ᾱ and
determines its magnitude and unit vector.
Solution
• The magnitude of vector Ᾱ = 2 x̂+ 3ŷ is

2
2
A  2  3  13
Unitvectoris

A 2 xˆ  3 yˆ
2
3
aˆ   

xˆ 
yˆ
13
13
13
A
Position and Distance Vectors
• A position vector is the vector from the origin
of the coordinate system O (0, 0, 0) to the
point P (x, y, z). It is shown as the vector R
• The position vectors can be written as:

R1  OP1  x1 xˆ  y1 yˆ  z1 zˆ

R2  OP2  x2 xˆ  y2 yˆ  z2 zˆ
A distance vector is defined
as displacement of a vector
from some initial point to a
final point. The distance
vector from P1 (x1, y1, z1) to
P2 (x2, y2, z2) is

 
R12  P1P2  R2  R1  x2  x1 xˆ   y2  y1 yˆ  z2  z1 zˆ
The distance between two vectors is:

d  R12 
x2  x1    y2  y1   z2  z1 
2
2
2
Example
Solution
Basic Laws of Vector Algebra
• Any number of vector quantities of the same
type (i.e. same units) can be cmbined by basic
vector operations.
• For instance, two vectors

A  Ax xˆ  Ay yˆ  Az zˆ
and

B  Bx xˆ  B y yˆ  Bz zˆ
• are given for vector operation below
Vector Addition and Subtraction
• Two vectors may be summed graphically by
applying parallelogram rules or head-to-tail
rule. Parallelogram rule draw both vectors
from a common origin and complete the
parallelogram however head-to-tail
rule is

obtained by placing vector B at the end of
vector Ᾱ to complete the triangle; either
method is easily extended to three or more
vectors.
Figure show addition of two vectors follow the rules, the
sum of the addition is
   
A B  B  A
 Ax xˆ  Ay yˆ  Az zˆ   Bx xˆ  By yˆ  Bz zˆ 
  Ax  Bx xˆ  Ay  By yˆ   Az  Bz zˆ
The rule for the subtraction of vectors follows easily from
that for addition, may be expressed
 
A B
as


A  B
The sign, or direction of the second vector is reversed
and this vector is then added to the first by the rule for
vector addition
 
 
  

A  B  A   B   Ax  Bx xˆ  Ay  By yˆ   Az  Bz zˆ
Example
Solution
Vector Multiplication
Simple product
• Simple product multiply vectors by scalars.
• The magnitude of the vector changes, but its
direction does not when the scalar is positive.
• It reverses direction when multiplied by a
negative scalar.


B  kA  kAaˆ  kAx xˆ  kAy yˆ  kAz zˆ
Dot Product
• Dot product also knows as scalar product.
• It is defined as theproduct of the magnitude of Ᾱ,
the magnitude of B, and the
 cosine of the smaller
angle, θAB between Ᾱ and B .
• If both vectors have common origin, the sign of
product is positive for the angle of 0⁰≤ θAB ≤90⁰.
• If the vector continued from tail of the vector, it
produce negative product since the angle is 90⁰≤
θAB ≤180⁰.
 
A  B  AB cos  AB
Commutative law :
A B  B  A
A B  B  A
Distribution law :
A  (B  C)  A  B  A  C
A  (B  C)  A  B  A  C
Associative law :
A  BC  D  ( A  B)(C  D)
A  BC  ( A  B)C
A  B  C  ( A  B)  C
A  ( B  C )  ( A  B)  C
• Consider two vector whose rectangular
 given such as
components are
A  Ax xˆ  Ay yˆ  Az zˆ
and

B  Bx xˆ  B y yˆ  Bz zˆ
• Therefore A  B yield sum of nine scalar
terms, each involving the dot product of two
unit vectors.
• Since the angle between two different unit
vectors of the rectangular coordinate system is
90⁰ (cos θAB =0) so
xˆ  yˆ  yˆ  xˆ  xˆ  zˆ  zˆ  xˆ  yˆ  zˆ  zˆ  yˆ  0
• The remaining three terms is unity because
unit vector dotted with itself since the
included angle is zero (cos θAB =1)
xˆ  xˆ  yˆ  yˆ  zˆ  zˆ
• Finally, the expression with no angle is
produced
 
A  B  Ax Bx  Ay By  Az Bz
Unit vector relationships
• It is frequently useful to resolve vectors into components
along the axial directions in terms of the unit vectors i, j, and k.
i  j  j  k  k i  0
i i  j  j  k  k  1
ii  j  j  k  k  0
i j  k
jk  i
k i  j
A  Ax i  Ay j  Az k
B  Bx i  B y j  Bz k
A  B  Ax Bx  Ay B y  Az Bz
i
A  B  Ax
Bx
j
Ay
By
k
Az
Bz
The Cross Product
Right Hand Rule
37
Example
Solution
Example
Solution
Scalar and Vector Triple Product
A B  C
Scalar triple product
The magnitude of A B  C is the volume of the parallelepiped with edges parallel to
A, B, and C.
A B
C
B
A
A  B  C  A  B  C  B  C  A  B  C  A  C  A  B  [ A, B, C ]
Vector triple product
A B  C
The vector A B is perpendicular to the plane of A and B. When the further vector
product with C is taken, the resulting vector must be perpendicular to A B and
hence in the plane of A and B :
( A  B)  C  mA  nB
where m and n are scalar constants to be determined.
C  ( A  B)  C  mC  A  nC  B  0
m  C  B
n  C  A
( A  B)  C   (C  B) A  (C  A) B
Since this equation is valid
for any vectors A, B, and C
Let A = i, B = C = j:
  1
( A  B)  C  ( A  C ) B  ( B  C ) A
A  ( B  C )  ( A  C ) B  ( A  B )C
A B
C
B
A
VECTOR REPRESENTATION: UNIT VECTORS
Rectangular Coordinate System
z
Unit Vector
Representation
for Rectangular
Coordinate
System
âz
â x
â y
y
x
The Unit Vectors imply :
â x
Points in the direction of increasing x
â y
Points in the direction of increasing y
âz
Points in the direction of increasing z
Example
Solution