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

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) 90o                                                      (b) 60o

(c) 120o                                                    (d) 0o

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

x2 = 2x2 (1 + cosa)

cos a = -1/2 = a = 120o                                                                             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 = 90o                                                                                   Answer is (c)

LAMI’S THEOREM

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