Vectors v/s Coordinate Geometry

Question

I know that vectors are very useful in Physics, but what is the purpose of vectors in mathematics, co-ordinate geometry is already a similar tool to tackle such kind of problems.

Answer 1

Here's one major reason to introduce vectors:

The key idea of calculus is to approximate a function locally by a linear function. But what does "linear" mean? This can't be answered (in a multivariable setting) without first introducing the idea of a vector.

(At least, describing a linear function without using the concept of vectors seems much less illuminating.)

Answer 2

Vectors spaces form one of the basic algebraic structures along with groups, rings and fields.

At an introductory level vectors create a concise vocabulary to generalize $\mathbb R^n$ (real numbers of n-dimensions). These are the "arrows" you see in physics class. However, physics rarely looks beyond $\mathbb R^3$ and most mathematicians would say, why stop at 3.

But, mathematicians use a more abstract definition of vector space, to include all objects that have a concept of addition and scalar multiplication. Polynomials are vectors. Functions can be treated as vectors.

Vectors and operations on vectors are the basic objects of linear algebra.

The fundamental operations of Calculus, differentiation and integration, can be treated as linear operators on the vector space of smooth functions.