Classical Physics - Vectors

Vectors

Vectors are those physical quantities which have magnitude and direction and are added according to triangle law of addition. in other words, directed segments which follow triangle law of addition are called vectors. 

Properties of a vector

In addition to magnitude and unit

It has specified direction 
    
It obeys triangle law of addition
    
Its addition is commutative that is, A + B = B + A
    
Its addition is associative that is, a + (B + C)

= (A + B) + C

Types of vectors vectors may be represented in two ways

Polar form in this form a = (r, θ) where r is magnitude and θ is the angle.

Cartesian form in this form A = a₁i +a₂J+a₃?

Where a₁, a₂ and a₃ are coefficients and i, J and ? are unit vectors along x, y, and z directions (axes) respectively

Null vector it is a vector whose magnitude is zero and hence directions becomes indeterminate.

Unit vector unit vector has magnitude as one and direction is specified. Unit vector a = A/|A|

That is, vector divided by its magnitude represents a unit vector.

Co-initial vectors if vectors have a common initial point they are said to be co-initial. For example, vectors Aand B shows in are co-initial.

Co-linear vectors like. Unlike, parallel, opposite or equal vectors may be grouped under a commonname co-linear vector, if they are either in the same line or parallel.

Co-planar vectors vectors lying in the same plane are said to be co-planar. 

Addition of vectors 

Vectors are added according to any of the three laws namely triangle law, parallelogram law and polygon law.

Triangle law of addition of vectors if two vectors acting on a body may be represented completely (in magnitude and direction) by two sides of a triangle taken in order then, their resultant is completely represented by the third side of the triangle taken in the opposite order. 

Thus, in (a) A = OP, B = PQ are represented by sides OP and PQ of a triangle, then their resultant OQ is taken in opposite sense.

That is, R = OQ or R = A + B

In other words, these vectors acting on a body may be represented completely by three sides of a triangle taken in order then the body is in equilibrium. 

We know A + B = R

Or A + B – R = R – R = O

Which is the basis of 2^nd definition? 

Parallelogram law if two vectors acting on a body simultaneously may be represented completely by two adjacent sides of a parallelogram then their resultant is represented completely by the diagonal of the parallelogram drawn from the common point.

Parallelogram law can be derived from triangle law.

In A = OP and B = OQ

Are two vectors represented by adjacent sides of a parallelogram OPLQ? Using equal vector property

PL = B (= OQ)

Now applying triangle law OP + PL = OL A + B = R.

Polygon law if (n – 1) vectors acting on a body may be represented completely by (n – 1) sides of an n – sided polygon taken in order then their resultant is completely represented by the closing side of the polygon taken in opposite order. 

Let A = OA, B = AB, C + BC and D = CD are the vectors acting on a body which are represented by the sides OA, AB, CD and DO of a polygon AOBCD in 

Then OD = OA + AB + BC + CD

Or R = A + B + C+ D

Proof according to triangle law 

OA + AB = OB

OB + BC = OC

OC + CD = OD

Rom (i), (ii) and (iii) we get OA + AB + BC + CD = OD

Triangle law (analytical proof) in 

AX = AP cos θ = B cos θ

PX = AP sin θ = B sin θ

OP²= OX ² + PX² = (OA + AX)² + PX²

=A² + B² cos² θ + 2AB cos θ - B² sin² θ

Or |R| = √A² + B² + 2 AB cosθ

Tan β = PX / OA + AX = B sin θ / A + B cos θ

R_max = A + B

That is, two vectors act in the same direction.

Rmin = A – B     (when θ = 180°)

That is, two vectors act in opposite direction. This gives a clue for all other cases

The magnitude of the resultant 

|A – B| ≤| R|≤|S + B 

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