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Vector Algebra and Vector Operator Review:
Vector Algebra:
If we walk 4km east and then walk 3km north we will have gone a total of 8km,
but we will be at a distance of 5km from our initial starting point. The arithmetic which
helps us describe such quantities is called vector algebra. In this review we will focus on
vector algebra in Cartesian coordinates. A spatial vector may therefore have up to 3
independent coordinates. Graphically we present vectors in the following way:
3 Dimensional
2 Dimensional
y
z
r
r
y
x
x
Traditionally, the notation used in many books is r  r  x  xˆ  y  yˆ  z  zˆ where the
unit vectors xˆ , yˆ and zˆ are sometimes denoted as iˆ, ˆj and kˆ, respectively.
Vector addition:
(1)
Vector addition is commutative: A + B = B + A
This is equivalent to saying that walking 4km east followed by walking 3km north is the
same as walking 3km north and then 4km east.
(2)
(A + B) + C = A + (B + C)
(3)
Subtraction: A – B = A + (-B)
(4)
Multiplication by a scalar/number: a(A + B)= aA + aB
We may add vectors in two manners, graphically and algebraically.
A  4 xˆ  3zˆ ; B  3xˆ  2 yˆ  3zˆ
Example 1:
C  A  B   4  3 xˆ  2 yˆ   3  3 zˆ  7 xˆ  2 yˆ  6 zˆ
D  B  A  (3  4) xˆ  2 yˆ  (3  3) zˆ   xˆ  2 yˆ
2
A  4 xˆ  2 yˆ ; B   xˆ  2 yˆ
Example 2:
y
y
(A+B)
y
A
B
A
(B+A)
A
B
x
x
(A-B)
-B
x
Dot product (‫)מכפלה סקלרית‬:
The dot product of two vectors is defined by:
(5)
A  B  AB cos
Where  is the the angle between the vectors when they are placed tale to tale and A and
B are the magnitude of A and B. The magnitude of a vector is given by:
(6)
A  Ax 2  Ay 2  Az 2
Clearly, the dot product of two vectors is a number (scalar). Algebraically, we use
xˆ  xˆ  yˆ  yˆ  zˆ  zˆ  1 ; xˆ  yˆ  yˆ  zˆ  xˆ  zˆ  0 
(7)
If we use A and B from example 2, we get
A  B  (4 xˆ  2 yˆ )  ( xˆ  2 yˆ )  4  (1)  2  2  0 .
When the dot product of two vectors is zero, we know that the two vectors are
perpendicular or orthogonal to each other. It is a straight forward matter to calculate the
angle between two vectors. We use the above result and compare it with (5). We get:
0  42  22 (1) 2  22 cos  10cos     , 3 as expected.
2
2
Similarly, if two vectors are parallel, (5) yields, A  B  AB , with the obvious result that
for every vector,
(8)
A  A  A2  Ax 2  Ay 2  Az 2
Example 3: law of cosines.
C=A-B
(9)
C 2  ( A  B)  ( A  B)  A2  B 2  2 AB cos
A
C

B
Cross product:
The cross product of two vectors is defined by
(10)
A  B  AB sin nˆ
where again,  is the angle between the vectors when they are placed tale to tale. The unit
vector n̂ is the unit vector in a direction perpendicular to the plane formed by the two
vectors. It follows from (10) that A  A  0 .
The cross product is neatly written down in component form as
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(11)
xˆ
A  B  Ax
Bx
yˆ
Ay
By
zˆ
Az
Bz
where we recall that the determinant of a 3x3 matrix is given by,
xˆ
yˆ
zˆ
Ax Ay Az  xˆ ( Ay Bz  Az By )  yˆ ( Az Bx  Ax Bz )  zˆ( Ax By  Ay Bz )
(12)
Bx By Bz
Unlike the dot product, the cross product is not commutative,
(13) ( A  B)  (B  A)
We may also have Triple products. These are simplified in the following way:
A  (B  C)  B  (C  A)  C  ( A  B)
(14)
A  (B  C)  B( A  C)  C( A  B)
(15)
*notice that B( A  C)  B  ( A  C)
Vector Operators:

Suppose we have a function of one variable, f(x). Differentiating f(x) with respect
to x gives us the rate of change of f(x) at x (or the slope of the function), f’(x).
When a function varies in more than one dimension, so does its rate of change,
which requires the use of a vector operator.
The Gradient:
The gradient of a function in Cartesian coordinates is defined by
T
T
T
xˆ 
yˆ 
zˆ
(16) T 
x
y
z
The Gradient, also denoted as “Grad”, is a vector, operating on the function T.


 
T   xˆ  yˆ  zˆ  T
y
z 
 x



yˆ  zˆ
(17)   xˆ 
x
y
z
The gradient operator has all the properties of a vector. It may be applied in 3 manners.
(a)
on a scalar function T: T  GradT (the Gradient)
(b)
on a vector V, via the dot product:  V  DivV
(the Divergence)
(c)
on a vector V, via the cross product:  V  CurlV
(the Curl)
The Divergence:
From the definition of the divergence, we get


 
  V   xˆ  yˆ  zˆ  (Vx xˆ  Vy yˆ  Vz zˆ)
y
z 
 x
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Vx Vy Vz


x
y
z
As for the geometrical interpretation of the Divergence, by looking at (18) we see that the
divergence is simply the sum of the relative change in each spatial dimension.
(18)
V 
The Curl:
The curl of a vector function is given by
xˆ
(19)
yˆ
V  
x
Vx
zˆ

y
Vy
 V V 
 xˆ  z  y  
z
z 
 y
Vz

 V V
yˆ  x  z
x
 z
 Vy Vx 

ˆ

z




y 

 x
Notice that the curl of a vector function is itself a vector function. We cannot take the curl
of a scalar, the same way we cannot take the divergence of a scalar – that’s meaningless.
As for the geometrical interpretation of the curl, as the name implies,  V is a measure
of how much the Vector function V “curls around” the point in question. This idea might
become clearer using the following example.
Example 5:
Suppose the functions F   yxˆ  xyˆ and H  yyˆ . Let’s calculate their curls:
xˆ
yˆ
zˆ

  zˆ   ( x)   ( y )   2 zˆ


x
y
z
y 
 x
y
x
0
Obtaining a rotational vector in the positive z direction means that the function rotates in
the x-y plane. For example, the rotation vector of earth around it axis points north. A
good way of visualizing the direction of the curl is to point the thumb of your right hand
at the direction of the rotation vector. The four fingers of your right hand will point at the
direction of the “curl”.
xˆ
yˆ
zˆ
F  
H  


0
x
y
z
0
y
0
The vector function H is a straight line, hence no curl.
Product Rules:
There are two ways to construct a scalar as the product of two functions:
(i)
fg
(product of two scalar functions)
(ii)
AB (dot product of two vector functions)
and two ways to make a vector:
(iii) fA
(scalar time a vector)
A  B (cross product of two vectors)
(iv)
Accordingly, there are six product rules. Two for gradients,
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(20) ( fg )  f g  gf
(21) ( A  B)  A  (  B)  B  (  A)  ( A  )B  (B  ) A
two for divergences,
(22)   ( fA)  f (  A)  A  (f )
(23)   ( A  B)  B  (  A)  A  (  B)
and two for curls,
(24)   ( fA)  f (  A)  A  (f )
(25)   ( A  B)  (B  ) A  ( A  )B  A(  B)  B(  A)
We nay also formulate the following three quotient rules:
 f  gf  f g
(26)    
g2
g
(27)
(28)
 A  g (  A)  A  (g )
  
g2
g
 A  g (  A)  A  (g )
  
g2
g
Second Derivatives:
By applying the gradient twice we can construct species of second derivatives.
1) Divergence of a gradient,   (T ) .


  T
T
T 
  (T )   xˆ  yˆ  zˆ 
xˆ 
yˆ 
zˆ 
y
z  x
y
z 
 x
 2T  2T  2T
(29)   (T )   2T  2  2  2
x
y
z
This operator is called the Laplacian of T. Notice that the Laplacian is a scalar. It is also
common to speak of a Laplacian of a vector. By this we mean
(30) 2V  2Vx xˆ  2Vy yˆ  2Vz zˆ
which simply means taking the Laplacian of each component separately.
2) curl of a gradient. The curl of a gradient is always zero.
(31)   (T )  0
3) the gradient of a divergence, (  V ) . Notice that the gradient of a divergence is not
the same as the Laplacian of a vector, 2V  (  )V  (  V) .
4) the divergence of a curl. The divergence of a curl, like the curl of a duivergence, is
always zero.
(32)   (  V )  0
5) the curl of a curl,
  (  V)  (  V)  2V
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Home Assignments:
1) What is wrong with the expression B  ( A  C) ?
x2  1 
1
2) Let T ( x, y, z )  3x 
z  
yx
x

a) T
b)   T
c) T
d)   H
e)   (  H )
f)   (  H)
2
3
2
3
x2  1
1 2

; H
xˆ  3x 2 yˆ   z   zˆ . Solve:
yx
x
