Numbers, Vectors, Matrices and Tensors

The direction from the origin to the point specified by the three coordinates represents a vector, having a length and a direction. Another example of a 3-vector is the velocity of a particle v, ,Vy,v, or alternatively dx/dt, dy/dt, dz/dt. For a system with N particles, the positions or velocities of all particles can be specified by a vector of length 3N, i.e. xi,yi,zi,X2,y2, .. y. z j or V uVyuV,uVy2,Vy2, , VyN,v,N.  

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley Sons, Ltd 

The notation for such vectors is often generalized to simply x = xiMr s. Xn- n), where N now refers to the total number of elements, i.e. equal to 3iV in the above notation. 

A complex number z can be interpreted as a 2-vector in an xy-coordinate system, z=x + iy, where i is the symbol for and x and y are real numbers. Here x and y are referred to as the real and imaginary parts of z. Alternatively, the complex number can be associated with polar coordinates, i.e. the distance r from the origin and the angle 0 relative to the real axis. 

The complex conjugate of a complex number z is denoted by z and is obtained by changing the sign of the imaginary part,i.e. z = x-iy or equivalently z = re . 

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