>>12532921Well, U looks to be complex numbers, and your i (imaginary unit) looks like a j. On the other hand, U_1 and U_2 look like scalars.
Complex numbers are vectors on a 2D plane with a multiplication rule. Ie. you can write a complex number c1 = a1 + i * b1 , where (a1, b1) are the 2 real-number coordinates of c1 as a point on the 2D plane. You can also write it as c1 = (cos(alpha) + i sin (alpha)) * r with (alpha, r) real numbers. This is the polar form of the complex number c1.
r is the length of the vector c1, and can be calculated as sqrt(a*a+b*b),
(cos(alpha) + i sin (alpha)) is a unit vector of the same direction as c where alpha is the angle angle c forms with the oX axis on the plane. The angle can be given by tan(alpha) = b / a. This comes from trigonometry (draw a right triangle with c = (a,b) , (0,0) and (a,0). ) .
The polar form of complex numbers is nice because it makes multiplication easy. If you have complex number c1 of polar form (a1, r1) and complex number c2 of polar form (a2, r2), then c1 * c2 has polar form (a1+a2, r1*r2). Basically, multiplication by a complex number of lenght r and angle alpha corresponds to a transformation that means: "scale by r and rotate by alpha counterclockwise" . This can explain the behavior of "i" . Multiplication of a vector by "i" means "rotate this vector 90 degrees" . That's why i^2 = -1. (it means rotation by 180 degrees) .
In your picture you have complex vectors as underlined shit, and scalars as non-underlined. This makes it hard as fuck to explain the equations in a typed medium. You have u as complex vector is the sum of a vector of length u1 and angle p1 with a vector of length u2 and angle p2. You're calculating u^2 as a scalar, which is the dot product <u,u> = <u1 + u2, u1 + u2> = <u1,u1> + <u2, u2> + 2<u1,u2>, where u1 and u2 refer to the vectors . Expanding yo the scalar forms you obtain the first equation enclosed in your paper.
