Vector analysis

Vector Analysis
Definition
A vector in n dimension is any set of n-components that transforms in the same manner as a displacement when you change coordinates
Displacement is the model for the behavior of all vectors
Roughly speaking: A vector is a quantity with both direction as well as magnitude. On the contrary, a scalar has no direction and remains unchanged when one changes the coordinates.
Notation: Bold face A, in handwriting A . The magnitude of the vector is denoted by A  A  A
Example:
Scalars: mass, charge, density, temperature
Vectors: velocity, acceleration, force, momentum

Vector Algebra
Vector Operations
(a) Addition of two vectors
Parallelogram law: To find A+B, place the tail of B at the head of A and

draw the vector from the tail of A to the head of B

B
A

A+B

From the definition, the addition of vectors is
(i) Commutative
A+B=B+A
(ii) Associative
(A+B)+C=A+(B+C)
(b) Negative of a vector
The negative of a vector is defined as the vector with the same magnitude but opposite direction

A
-A

(c) Multiplication by a scalar
Multiplication by a positive real number a multiplies the magnitude by a times while leaving the direction unchanged.
A
A aA aA

Multiplication by 0 gives the null vector 0 which satisfies
A+0=A
Multiplication by a negative real number a is defined by aA=|a|(-A) A
A
aA
|a|A
Multiplication is distributive, i.e., a(A+B)=aA+aB (d) Subtraction of two vectors
A-B is defined by A+(-B)
(e) Dot product (scalar product, inner product) of two vectors
A B  AB cos 

A


B
Dot product is commutative:
A BB A

Dot product is distributive:
A  B  C  A B  A C

(Can you prove it?)
If A, B perpendicular,    / 2  A B  0

If A, B point to the same direction,   0  A B  AB
In particular, A A  A2
(f) Cross product (vector product, outer product) of two vectors
ˆ
A  B  AB sin  n

ˆ n B


A
ˆ
n is a unit vector ( magnitude =1 )