Scalars and Vectors

PHYSICAL QUANTITY: SCALAR and VECTOR

A physical quantity is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as the combination of a magnitude and a unit.

Ex: 5 kg in this ‘5’ is magnitude and ‘kg’ is unit.

Physical quantity can be operated in two parts:

            (i) Scalar Quantity               (ii) Vector Quantity

Scalars:

• A physical quantity which has only magnitude.

• They do not have any direction.

  ex: time, mass, speed, distance, volume, energy, work, temperature, electric charge.  

Vectors:

• A vector quantity is defined as the physical quantity that has both, magnitude as well as direction.

ex: displacement, force, torque, momentum, acceleration, velocity, etc.

Representation of Vector:

A vector is represented as a directed line segment.

                  this vector can be represented as AB乛 or ā.

•                Magnitude of this vector is represented as |AB|  or  |ā|  or ā.
•            Magnitude is ‘the amount’ or ‘quantity of something’.
•            Here magnitude is distance(d) between point A and B.

Types of Vectors:

Zero Vector or Null Vector:  A vector which has zero magnitude i.e. having same initial and terminal point. ex- point(dot) in space.

Unit Vector: A unit vector is a vector having a magnitude of unity or 1 unit. A unit vector in the direction of a given vector a is denoted as ậ.

￫ unit vector in the direction of a  is: ậ = ā/|ā|
ex: Unit vectors along x-axis, y-axis and z-axis are  î , ĵ and k̂  respectively.

Magnitude of a Vector:

If  Ā = x î + y ĵ + z k̂ is,

                            then magnitude of Ā is :

                                                       |Ā|= √x²+y²+z²

Q. Find the magnitude of Ā = 5 î - 2 ĵ + 7 k̂ .

Ans.  Magnitude of the given vector is

                                    |Ā|= √(5)² + (-2)² + (7)²

                                    |Ā|= √78

       Q. Find the unit vector in the direction of A = 4 î + 3 ĵ - 5 k̂ .
           Ans.                           Â = Ā/|Ā|
                                        here, |Ā|= √(4)²  + (3)²  +(-5)² 
                                                   |Ā|= √50
                                                   |Ā|= 5√2

              so, unit vector along Ā = 1/5√2(4 î + 3 ĵ - 5 k̂)

Position Vector: 
              Position vector simply denotes the position of a point in the three-dimensional Cartesian system with respect to a reference origin.
              If P is taken as reference origin and P is any arbitrary point in space then the vector 
OP−→− is called as the position vector of the point.

Co-initial Vector:  Two or more vectors having same initial point are known as Co-initial vectors.

Co-Planer Vector: Three or more vectors lying in the same plane are known as Co-Planer vectors. 

Like and Unlike Vectors: Vectors having the same directions are known as like vectors and vectors having the opposite direction are known as unlike vectors.

Co-Linear Vector: Vectors which lie along the same line or parallel lines are known to be co-linear vectors. They also known as parallel vectors.

Equal Vector:  Two or more vector are said to be equal if all  have same magnitude and direction. They can have different initial points.

Negative of a Vector:  Consider a vector has a magnitude and direction. A vector is said to be negative of a vector if it has same magnitude but opposite direction. 

Displacement Vector: If a point is displaced from position A to B then the displacement AB represents a vector AB which is known as the displacement vector.