1) The document defines key vector concepts including position vectors, vector products, scalar products, and their properties and notation. 2) It also introduces gradient, divergence, curl - vector operators that describe how quantities change in space. 3) The Laplace operator, denoted by ∇2, is defined as the divergence of the gradient, and relates to Laplace's equation. It can also be applied to vector fields.

2. •A vector – standard
notation for three
dimensions.
•Magnitude of a vector.
•Position vector is a
vector r from the origin to
the current position. where
x,y,z, are projections of r
to the coordinate axes.
kAjAiAAAAA zyxzyx

 ),,(
222
zyx AAAAA 

kzjyixzyxr

 ),,(

3. Vector product (cross
product) – is defined as
Where Θ is the smaller angle
between vectors
a and b and n is unit vector
perpendicular to the
plane containing a and b.
Component Notation-
nabba

 sin.
sin abbaA

kbabajbabaibaba
bbb
aaa
kji
bac
xyyxzxxzyzzy
zyx
zyx



)()()( 

Basic Vector Properties Of
Vector Product
0


baba
abbaba
abba




4. Scalar product (dot product) – is defined as
Where Θ is a smaller angle between vectors
a and b and S is a resulting scalar. Sbaba
Sbaba
i
n
i
i 

1
cos



For three component
vectors can be written as- cos abbabababaS zzyyxx

Geometric interpretation –
scalar product is equal to the
area of rectangle having a and
b . Cos Θ as sides. Blue and
red arrows represent
original vectors a and basic
properties of the scalar
product. abbaba
baba
abba






0

5. )()()( baccabcba


)()()( bacacbcba


V
ccc
bbb
aaa
cba
zyx
zyx
zyx
 )(


6. The gradient of a scalar field is a vector field that points in the
direction of the greatest rate of increase of the scalar field, and
whose magnitude is that rate of increase.
z
S
k
y
S
j
x
S
iSgradS











7. Gradient
(Nabla operator)
Divergence
Curl
z
S
k
y
S
j
x
S
iSSgrad










z
A
k
y
A
j
x
A
iAAdiv zyx




















































y
A
x
A
k
x
A
z
A
j
z
A
y
A
i
AAA
zyx
kji
AAcurl
xyzx
yz
zyx





8. This expression occurs so often that
we abbreviate it as -
2
Ñ = Ñ×Ñ
The operator is called the Laplace
operator due to its relation to Laplace’s
equation . The equation is -
2 2 2
2
2 2 2
0
f f f
f
x y z
¶ ¶ ¶
Ñ = + + =
¶ ¶ ¶

9. We can also apply the Laplace operator
to a vector field
F = P i+ Q j + R k
In terms of its components -
2 2 2 2
P Q RÑ = Ñ +Ñ +ÑF i j k