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VECTOR CALCULUS

Subhalakshmi Lamba

Vector Product

CROSS PRODUCT

Vector Product (contd.)

Vector Product (contd.)

?

- Magnitude

Vector Product (contd.)

?

Vector Product (contd.)

?

- Direction A rule is required !!

Right-Hand Rule

- The order of vector multiplication
- is important.

Geometrical Interpretation

Properties

- Vector multiplication is not
- Commutative.

- Vector multiplication is Distributive

- Multiplication by a scalar

Properties(contd.)

Properties(contd.)

Angle between them 0º

Angle between them 90º

Vector Product Components

?

Vector Product Components

Vector Product Components

Vector Product Determinant

Examples in Physics

The torque produced by a force is

Examples in Physics (contd)

The angular momentum of a particle with

respect to O

- Examples in Physics (contd)

The force acting on a charged particle

moving in a magnetic field,

Positive charge

Negative charge

- Applications

- A simple cross product

- An Example

C H E C K

- Applications(contd.)

- finding a unit vector
- perpendicular to a plane.

- An Example

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.

- TRIPLE PRODUCTS

- Scalar Triple Product

- Scalar Triple Product (contd.)

- Scalar Triple Product (contd.)

- Scalar Triple Product (contd.)

Area of the base

- Scalar Triple Product (contd.)

Volume of the parallelepiped

Properties

PSUEDOSCALAR

Properties

- If any two vectors of the scalar triple product

are equal, the scalar triple - product is zero.

- Vector Triple Product

An Example

An Example

An Example

What is the force that q1 exerts on q2 ?

v1

v2

B

q1

q2

r

An Example

v1

v2

B

q1

q2

r

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.

SUMMARY

SUMMARY

- Vector Triple Product

- Quadruple Product of vectors