What is the correct way to represent vectors in mathematics?

Question

What is the correct way to represent vectors in mathematics? Mainly between component methods and angle and magnitude methods.

Answer 1

Consider a vector space $V$ simply as an algebraic structure. You do not have the notion of an angle and length, this is only possible if you equip your vector space with an additional structure called an inner product, i.e. the standard inner product on $\mathbb{R}^n$ defined by $$\langle x,y\rangle := \sum_{i = 1}^n x_i y_i$$ for $x,y \in \mathbb{R}^n$. So in mathematics, you just consider vectors as elements belonging to an algebraic structure. I mean, how would you represent for example polynomials? This goes over the common understanding of vectors simply as elements of $\mathbb{R}^n$. If you have equipped your space with an inner product it doesn't really matter which method you're using.

Answer 2

Clearly, the correct way is the component representation. Because the addition formulas are much easier:

$$s_x=a_x+b_x,s_y=a_y+b_y,$$ vs. the terrible

$$s_r=\sqrt{(a_r\cos a_\theta+b_r\cos b_\theta)^2+(a_r\sin a_\theta+b_r\sin b_\theta)^2},\\ s_\theta=\arctan\frac{a_r\sin a_\theta+b_r\sin b_\theta}{a_r\cos a_\theta+b_r\cos b_\theta}.$$

And clearly, the angle/magnitude representation is by far the best as it gives an elementary rotation formula

$$b_\theta=a_\theta+\phi$$ vs. the unhandy

$$b_x=a_x\cos\phi-a_y\sin\phi,\\ b_y=a_x\sin\phi+a_y\cos\phi.$$