Vector parallelogram theorem

         If two given incline vectors can be represented by two adjecent line of a parallelogram.(In a diverging manner)the result is given by diagonal after completing the parallelogram.

Deriving the vector parallelogram theorem eqation

Pythagoras theorem OBDΔ

      OD²+DB² = OB²

      

      OD = OA+AD

  

    

  cosθ = AD                             sinθ = BD                                 

              AB                                        AB

      AD = ABcosθ                          BD = Asinθ

            = Qcosθ                                =  Qsinθ

  

  OA = P 

  OD = P+Qcosθ                       BD = Qsinθ

  OB² = R² = (P+Qcosθ)² + (Qsinθ)²

                =  p² + 2PQcosθ + Q²sin²θ + Q²cos²θ

                =  P² + 2PQcosθ + Q²(sin²θ + cos²θ)

                = P² + 2PQcosθ + Q²

  tanα = BD  =  Qsinθ     

          = OD  =  P+Qcosθ

Resultant of incline vectors

   If two vectors or a vector system is given to find the resultant it is not simple as finding the resultant of parallel vectors.

Triangular law 

If given two vectors can be represented by the adjacent lines of a triangle, the magnitude of the resultant vector is given by the line which complete the triangle. And the direction is the opposite flow direction of the two vectors.

Polygonture's Law

    If several vectors can be represented by the adjacent lines of a polygon. The magnitude of the resultant is given by the line which complete the polygon and the direction in the opposite flow direction of the other vectors.

a+b+c+d+e = f

Apply triangular law for each and every triangle.

If three vectors can be represented by a complete triangle the resultant would be 0

R=0

                 

Resultant

Resultant of parallel vectors

            Resultant is the single force which have the same effect of two or several forces .If two vectors are in the same direction the resultant magnitude can be obtained by the addition of two magnitude of vectors. And the direction is same as the two vectors.(img:7)

img:7

If two vectors are in the opposite direction the magnitude of the resultant can be obtained by the subtraction of the magnitude of the two vectors.the direction of the vector is the direction of the bigger vector. (img:8)

img:8

The maximum resultant of two vectors can be obtained when the two vectors are parallel to each other and have the same direction. All the other states are between these two values.(img:9)

img:9

R is any resultant of p and q (incline state)

R(min) < R < R(max)

The magnitude of a vector is devoted by the modules.(img:10)

img:10

Introduction to mechanics

Mechanics

Introduction to physics

What is known as vectors?

 vectors are quantities which have a magnitude and a direction.                                                                        eg:-  velocity

                  Force

                  Momentum                      

                  Displacement

                  Acceleration

what is known as scalers

scalers are quantities which only have a magnitude.
         eg:-  power 
                 distance
                 speed
                 pressure

properties of vectors

   A vector can be multiplied by a scaler(img:1). If the scaler is positive it wan't change the direction of the vector. But if the scaler is negative then it will change the direction as well(img:2). But in both cases it will change the magnitude of the vector.

img:1

img:2

A vector can be represented by two ways.

              1. we can use a letter and underline then it represents a whole vector(img:3).

img:3

             2. we can use capital letters for the ends of vectors and draw an arrow necessarily to indicate the direction head(img:4).

img:4

Negative vectors 

         Negative vectors are vectors which have the opposite direction of the current vector.(img:5) (img:6)

img:5

img:6