2.Mathematical methods

 Scalar Quantities

Scalar quantities are quantities that are described only by a magnitude.

Vector Quantities. 

Vector quantities have both magnitude 

and a direction of action.

 

Types of Vector

1) Zero vector (Null vector): A vector

having zero magnitude with a particular

direction (arbitrary) is called zero vector.

2)Resultant vector: The resultant of two or more vectors is that single vector, which  

produces the same effect, as produced by

all the vectors together.

3)Negative vector (opposite vector): A

negative vector of a given vector is a

vector of the same magnitude but opposite

in direction to that of the given vector.

4)Equal vector: Two vectors A and B 

representing same physical quantity are

said to be equal if and only if they have  

the same magnitude and direction.

5)Position vector: A vector which gives

the position of a particle at a point with

respect to the origin of a chosen co-

ordinate system is called the position

vector of the particle.

6)Unit vector: A vector having unit

magnitude in a given direction is called

a unit vector in that direction. 

Scalar Product (Dot Product):

The scalar product or dot product of two

nonzero vectors P^and Q^ is defined as the  

product of magnitudes of the two vectors and the cosine of the angle θ between the two vectors

Characteristics of scalar product

(1) The scalar product of two vectors is

equivalent to the product of magnitude of one vector with the magnitude of the component of the other vector in the direction of the first.

(2) Scalar product obeys the commutative law of vector multiplication.

(3) Scalar product obeys the distributive law of multiplication

Vector Product (cross product):

The vector product or cross product of two

vectors ( P and Q ) is a vector whose magnitude is equal to the product of magnitudes of the two vectors and sine of the smaller angle (θ) between the two vectors.

Characteristics of Vector Product:

(1) Vector product does not obey commutative law of multiplication.

(2) The vector product obeys the 

distributive law of multiplication.

Scalar and vector products are very

useful in physics. They make mathematical formulae and their derivation very elegant.