Begin2.DVI

been represented by a variety of notations. Some of the more common notations

employed in various textbooks to denote rectangular unit base vectors are

ˆı, ˆ, ˆk, ˆex, ˆey, ˆez, ˆı 1 ,ˆı 2 , ˆı 3 , (^1) x, (^1) y, (^1) z, ˆe 1 , ˆe 2 , ˆe 3

The notation ˆe 1 , ˆe 2 , ˆe 3 to represent the unit base vectors in the direction of the

x, y, z axes will be used in the discussions that follow as this notation makes it easier

to generalize vector concepts to n-dimensional spaces.

Scalar or Dot Product (inner product)

The scalar or dot product of two vectors is sometimes referred to as an inner

product of vectors.

Definition (Dot product) The scalar or dot product of two vectors A
 and B


is denoted

A
 ·B
 =|A
||B
|cos θ, (6 .8)

and represents the magnitude of A
 times the magnitude 
B times the cosine of θ,

where θis the angle between the vectors 
Aand B
 when their origins are made to

coincide.

The angle between any two of the orthogonal unit base vectors ˆe 1 ,ˆe 2 ,ˆe 3 in

cartesian coordinates is 90 ◦or π 2 radians. Using the results cos π 2 = 0 and cos 0 = 1 ,

there results the following dot product relations for these unit vectors

ˆe 1 ·ˆe 1 = 1

ˆe 1 ·ˆe 2 = 0

ˆe 1 ·ˆe 3 = 0

ˆe 2 ·ˆe 1 = 0

ˆe 2 ·ˆe 2 = 1

ˆe 2 ·ˆe 3 = 0

ˆe 3 ·ˆe 1 = 0

ˆe 3 ·ˆe 2 = 0

ˆe 3 ·ˆe 3 = 1

(6 .9)

Using an index notation the above dot products can be expressed ˆei·eˆj=δij

where the subscripts iand jcan take on any of the integer values 1 , 2 , 3 .Here δij is

the Kronecker delta symbol^1 defined by δij =

{ 1 , i =j

0 , i 
=j

.

The dot product satisfies the following properties

Commutative law A
·B
 =B
 ·A


Distributive law A
·(B
 +C
) = A
·B
+A
·C


Magnitude squared A
·A
=A^2 =|A
|^2

which are proved using the definition of a dot product.

(^1) Leopold Kronecker (1823-1891) A German mathematician.