Chapter 2 : Vectors and Scalars

Question 1 : Define Scalar and Vector Quantities. Also show the representation of vectors.

Answer : 

Scalar Quantities : Those physical quantities which have only magnitude is called scalar quantities. For example : Mass, Length, Time, Temperature, Speed, Volume, Distance, e.t.c.

Vector Quantities : Those physical quantities which have both magnitude and directions are called vector quantities. For example : Velocity, Displacement, Acceleration, Force, Magnitude Field, e.t.c.

Representation of vectors :

A vector is represented by a letter with an arrow over it. A vector is represented by A, V, e.t.c.

Geometrically, Vector is represented by a straight line with an arrow head. 

Question 2 : Difference between Vectors and Scalars

Answer : The major difference between vectors and scalars are as follows :

Vectors	Scalars
1. Vectors have both magnitude and directions.	1. Scalars have only magnitude.
2. Vectors are denoted by letters with an arrow head over it.	2. Scalars are denoted by numbers with specified units.
3. Vectors are added or subtracted vectorically or geometrically.	3. Scalars are added or subtracted algebraically.
4. Crossed product of vector is a vector.	4. Product of scalar is scalar.
5. It does not obey commutative law.	5. It obey commutative law.
6. The crossed product of a vector gives its a zero value	6. The dot product of a scalar gives a square of its magnitude.

Question 3 : Describe Resolution of Vectors.

Answer : The process of splitting of a vector into two or more than two vectors is called resolution of vectors. The splinted or resolved parts are called component of given vectors. The total sum of all component of a vectors is equal to the original vector.

The vector component along x-axis is called x-component or horizontal component and along y-axis is called y component or vertical component.

For this, In right angled triangle OAC, 

SinƟ = AC/OC

or, OCSinƟ = AC

or, RSinƟ = AC

i.e. AC=OB=RSinƟ=Y-component

 And,

CosƟ = OA/OC

or, OCCosƟ = OA

or, RCosƟ = OA

or, OA= RcosƟ=X-component

Note :

 1) If  Ɵ=90° then AC=OB=Sin90°, i.e.OB=R and OA=RCos90°=0

2) If  Ɵ=0° then AC=OB=RSin0°=0 and OA=RCos0°=R      

 

 

Question 4 : Describe Polygon and Parallelogram Law of Vector Edition.

Answer :

Polygon Law of Vector Edition :

 It states that,"If a number of vectors be represented both in magnitude and direction by the side of polygon taken in same order then the resultant is represented completely in magnitude and direction by closing side of the polygon taken in opposite order."

Parallelogram of Vector Addition : 

 This law states that,"If two vector quantity simultaneously acting on a body are represent in both magnitude and direction by the two adjacent sides drawn at a point of parallelogram then the resultant vector is represented both in magnitude and directions by the diagonal passing through same point."

Question 5 : At what angle the magnitude of resultant of two equal vector is equal to magnitude of either one.

Solution:

Let two vectors P and Q and their magnitude is equal to each other (P=Q=x).

Again, R=x

Then,

R=√P²+Q²+2PQCosƟ

Or, x=√P²+Q²+2PQCosƟ

Squaring both side,

(x)²=(√P²+Q²+2PQCosƟ)²

Or, x²=2x²+2x²CosƟ

Or, , x²=2x²(1+CosƟ)

Or, 1/2= 1+CosƟ

Or, 1/2-1 = CosƟ       

Or, 1-2/2 = CosƟ

Or, CosƟ = -1/2

Or, CosƟ = Cos120°

Thus, Ɵ=120°