SCALAR & VECTOR

Most of our physical quantities are linearly depends its variables. I not sure to explain what is mean by linear dependence, suppose we have a physical quantity R is the function of variables x, y and z if they linearly depends R can be written as,

                            R = c₁x + c₂y + c₃z  where c₁, c₃, and c₄ are constants or values. The sign “+” is not only ordinary addition also a symbol of linear operator; avoiding this confusion R can be written as the matrix form

.R = [c₁  c₂  c₃]   [x

                              y

                              z]                         

                                                                                                                                                                 In physics most of the quantities can be written as the linear combination of other physical quantities either fundamental or non fundamental quantities. But in our undergraduate levels, we deals with majority the linear combination of constants or direction or both, these are respectively called scalar and vector.

Vector is the quantity has both magnitude and direction this is we learned, but please tells like that, vector which is the quantity gives the direction, for example velocity of water gives the direction of water flows. Not, velocity of fluid is in the direction of motion. And don’t distinguish scalar and vector in the scale of magnitude and direction. They are differs in the order of writing the linear combination, scalar is zero order matrix that is no specific indication, vector is the first order matrix that is only one indices is needed. Let a_i is the vector and ‘i’ is the indices it has values, i = 1,2,3,…….etc.. Due to the single indices vector is either row only or column only matrix. A_ij  is not a vector because it has two indices ‘i’ and ‘j’.

From above discussion you ask that what is the significance of indices, this is why the mathematics and physics differ. The  scalar and vector are the mathematical concepts, but the indices gives the physical significance, here indices in vector gives the space coordinates we know that each coordinates in a coordinate system is independent so only one value is possible In each coordinate, this why the vector has only one indices.Fundamental levels we can discuss higher order (more than one indice) matrix or tensors.