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

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    I think the correct answer is that it depends on what you're trying to say or do. Depends on the context. – 2017-02-03
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    Any way you think that makes what you are saying clear. – 2017-02-03
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    This isn't answering my question. I need expert testimony! – 2017-02-03
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    magnitude and angle or magnitude in each direction. – 2017-02-03
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    As with any mathematical object, it doesn't matter how you specify a vector, as long as the specification is understood and is precise enough to pin it down completely. Specifying a vector as a list of coordinates (i.e. components) with respect to some coordinate system is one way, and specifying it by choosing a direction and magnitude is another way. You could also specify it by defining its start and end points. No one of these methods is more or less correct than any of the others, as they all uniquely define the vector in question. – 2017-02-03
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    There are various ways of parametrising a point/vector. None are more correct than others. – 2017-02-03

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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.

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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

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