All books or courses have separate chapters or courses for vector,coordinate geometry and complex numbers.But Vector,coordinate geometry and complex numbers are closely related and can be integrated beautifully to be taught together for a better understanding.But why are these not taught together in an integrated way?Are there any concepts in one topic that is not included in the other which makes the teaching of the three topics separetely necessary?Or are these the same thing being taught in different ways?
Thanks for any help!!
They are related, but not the same. For instance, while the complex numbers have some similar geometric properties as the plane, they also have additional structure: you can multiply and divide complex numbers in a non-ambiguous way, whereas this is not possible for vectors on the plane. $\mathbb{R}^2$ also has a different differential structure than $\mathbb{C}$. Every complex-differentiable function is not only also infinitely differentiable on its domain, it is equal to its Taylor series everywhere on its domain. On the other hand, there are plenty of examples of functions on $\mathbb{R}^2$ that are only once differentiable.
Now, these fields are certainly related, and when studying the theory of complex variables one sometimes makes use of the underlying vector structure, and some basic knowledge of the algebraic properties of $\mathbb{C}$ is convenient when studying linear algebra, but they are separate enough as ideas that the modern teaching style makes sense. Further, combining these topics runs the risk of having a student believe that vectors are an inherently two-dimensional class of objects.