Vector and Scalars

VECTOR AND SCALARS

Physical Quantity

May have numerical value, units and any specified direction.

Note:-  Some physical quantities only having numerical value not specified direction.

Ex:-  Refractive index, strain etc.

We can say any physical quantity Must/may have numerical value(n), unit (u) and specified direction.

PHYSICAL QUANTITY (N + U + DIRECTION) MAY/MUST HAVE

Note:

Scalars or zero order tensors: Any physical quantity have only one component.

Vectors or first order tensors: Any physical Quantity have component greater then one but less than or equal to four.

REPRESENTATION OF VECTOR:

Types of vectors

(1) Polar Vectors:

Vectors related to linear motion of any object Ex. Displacement, velocity, Force etc.

(2) Axial vectors:

Vector represent rotational effect and are always along the axis of rotation Ex: Angular velocity, torque, angular momentum etc.

(3)  Null vector or zero vectors:

Vector whose magnitude is zero and direction in determinant.

(4) Unit vectors:

Vector having magnitude equals to one (unity) but must be some specified direction representation of unit vector – , A  

 

STANDARD UNIT VECTOR

These are

Direction should be on your copy

UNDERSTANDING OF UNIT VECTORS

1 –D Motion: Motion along x or y or z

A car is moving with 60 k/m toward or due east then speed (scalar) = 60 km/h

Velocity (Vector) = 60 km/h

2 –D Motion:

A car is moving with 60 k/m due north – east then

 

 

2 –D vector

 

With x axis or with horizontal

 

With Y-axis or with vertical

3 –D Vector

 DIRECTION

 

With x-axis

 

With y-axis

 

With z-axis

ADDITION AND SUBSTRACTION OF VECTORS

Law of parallelogram

 

 

 

 

 

 

 

 

 

 

 

 

            RESOLVING OF ANY VECTOR

Example. Two equal vector have a resultant equal to either of the two. The angle between them is                                                                                                                    

(a) 90^o                                                      (b) 60^o

(c) 120^o                                                    (d) 0^o

Solution: By using expression R² = P² + Q² + 2PQ cosa

 

x² = 2x² (1 + cosa)

cos a = -1/2 = a = 120^o                                                                             Answer is (c)

Example: Two vector having equal magnitude of x units acting at an angle of 45⁰ have resultant   units the value of x is                                                             

(a) 0                                        (b) 1

(c)                                       (d)

Solution: Using the expression R² = P² + Q² + 2PQ cosµ

 

 

 

 

 

Þ x² =1                                                                    Ans (b) Þ x = 1 Ans (b)

 

 

(a) 0°                                                   (b) 180°

(c) 90°                                                 (d) 120°

 

 

= 4PQ cos q = 0

   Cos q = 0

   q = 90^o                                                                                   Answer is (c)

          LAMI’S THEOREM

        

  

   

 

Note: Vector triple product is not in our Physics syllabus.                                                                                                                                                                                                                   

SCALAR OR DOT PRODUCT

   Product of magnitude of two Vectors with cosine of angle between them.

     

    

    

   For perpendicular Vectors: ,q = 90° and cos 90° = 0

              

    

             

     

 Dot product of Standard unit Vectors:

                                                                                                     

                                                                                                        

        

  are mutually perpendicular then

the value of m is

      (a) - 2

      (b) - 3

      (c) + 3

      (d) + 2

            

          

  2 + m + 1 = 0

  m = - 3                                                                                           Answer (b)  

  Method to find dot or Scalar product

   

   

   

  Finding Scalar and Vector Component (projection)

     

       

          

    

  

  

         

  

       

   

   

   

  

   

   

   

   

                     

                      

   

  

  Component (projection) in the direction and in the perpendicular direction

   

   

   

   

   

   

   

   

              

           vector or cross product   

    Product of magnitude of two Vectors with sine of angle between them.

    

    

   For perpendicular Vectors:

    

   For parallel Vectors:

    

    

    

   For antiparallel Vectors:

    

    

    

   Cross or Vector product of Standard unit vectors

    

    

   USING RIGHT HAND SCREW RULE

   

   Method to find cross or vector product

    

    

    

    

          AREA OF TRIANGLE 

    

   Area of parallelogram:

    

         DIAGONAL OF PARALLELOGRAM

   Example If the diagonals of a parallelogram are and  ,then area of this

   parallogram will be

     (a) 2 unit²                                        (b) 3 unit²

     (c) 4 unit²                                        (d) 1 unit²

    

    

    

    

    

   = 2 unit²                                                         Answer (a)

   Example: Unit vector perpendicular to two vectors Finding a unit vector

   perpendicular to

    

    

    

    

    

     

    

    

    

   Example: Consider three Vectors

   Scalar triple product

    

    

    

    

   

    (a) 0

    (b) 6

    (c) 12

    (d) 18

    

    

   = 2 (3 n-2) -3 (15 + 1) -2 (10 + n) = 0

   4n – 72 = 0

   n = 18                                                         Answer (d)