VECTOR CALCULUS

1
VECTOR CALCULUS
Subhalakshmi Lamba
2
Vector Product
CROSS PRODUCT
3
Vector Product (contd.)
4
Vector Product (contd.)
?

• Magnitude

5
Vector Product (contd.)
?
6
Vector Product (contd.)
?

• Direction A rule is required !!

7
Right-Hand Rule
8

• The order of vector multiplication
• is important.

9
Geometrical Interpretation
10
Properties

• Vector multiplication is not
• Commutative.

• Vector multiplication is Distributive

• Multiplication by a scalar

11
Properties(contd.)
12
Properties(contd.)
Angle between them 0º
Angle between them 90º
13
Vector Product Components
?
14
Vector Product Components
15
Vector Product Components
16
Vector Product Determinant
17
Examples in Physics
The torque produced by a force is
18
Examples in Physics (contd)
The angular momentum of a particle with
respect to O
19

• Examples in Physics (contd)

The force acting on a charged particle
moving in a magnetic field,
Positive charge
Negative charge
20

• Applications

• A simple cross product

21

• An Example

C H E C K
22

• Applications(contd.)

• finding a unit vector
• perpendicular to a plane.

23

• An Example

24
SUMMARY

• Magnitude and Direction of a vector remain
  invariant under transformation of coordinates.

• Product of a vector with a scalar

• Vector product directional property,
• denotes an area.

25

• TRIPLE PRODUCTS

26

• Scalar Triple Product

27

• Scalar Triple Product (contd.)

28

• Scalar Triple Product (contd.)

29

• Scalar Triple Product (contd.)

Area of the base
30

• Scalar Triple Product (contd.)

Volume of the parallelepiped
31
Properties
PSUEDOSCALAR
32
Properties

• If any two vectors of the scalar triple product
  are equal, the scalar triple
• product is zero.

33

• Vector Triple Product

34
An Example
35
An Example
36
An Example
What is the force that q1 exerts on q2 ?
v1
v2
B
q1
q2
r
37
An Example
v1
v2
B
q1
q2
r
38
SUMMARY

• A physical quantity which has
• both a magnitude and a direction
• is represented by a vector

• A geometrical representation

• An analytical description components

• Can be resolved into components along any three
  directions which are non planar.

39
SUMMARY
40
SUMMARY

• Vector Triple Product

• Quadruple Product of vectors