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I'm only 4 weeks into my first college-level physics course, so this is a very elementary question.

Also, in asking this question, I'm just playing devil's advocate in order to better understand the historical motivation for the use of vectors to describe physical phenomena. I'm not actually proposing we stop using them. :) So, please don't bash me for asking this innocent question.

As I was reading my textbook today, I wondered why vectors were invented and are still useful today. I have a little experience with them from my Trigonometry course, but they've always seemed a bit counterintuitive to me, as a tool for modeling and calculating.

What are the benefits of representing magnitude and direction (or other related values) with a single variable, instead of treating them as separate values with distinct variables?

What are the drawbacks (aside from increased complexity), if any, of using vectors instead of performing calculations and building models in which the speed and direction of a moving object (for example) are treated as distinct values with a distinct variable for each quantity?

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  • $\begingroup$ Try adding two vectors! It's easy: in pictures, you just put them head to tail. In Cartesian components you just add the components. Choosing magnitude and direction essentially ruins the picture approach; it's basically committing yourself to a fixed coordinate system (spherical). $\endgroup$ – knzhou Feb 7 '16 at 0:49
  • $\begingroup$ It is not only about the "picture" approach: it is much easier to algebraically add the components of a vector in some coordinate system rather than doing the analogous manipulation to magnitudes and directions. $\endgroup$ – Giorgio Comitini Feb 7 '16 at 1:46
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Vectors and vector notation can tend to simplify equations and algebra. They might feel like they just complicate things at first, but once you get used to them they can provide an intuitive way to simplify calculations and concepts. Consider for example Newton's second law. While without vectors we'd have to write $$F_x = m \times a_x$$ $$F_y = m \times a_y$$ $$F_z = m \times a_z$$ With vectors, this is $$\vec{F} = m \times \vec{a}$$ Which is much easier to write. Another example would be once you start having to use cross products in equations. Not only do they combine several equations into one, but they also make that one equation shorter than any of the three you'd have to write without vectors.

These types of abstraction are common in physics. They might be counter-intuitive at first, but in the long run they can help you calculate and think faster.

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  • $\begingroup$ This is a great answer, as is the one from @ACuriousMind. When I decided to ask the question, I was thinking in terms of linear and two-dimensional motion (since that's what we're studying now), but (in addition to the reasons you gave) the more dimensions you add, the benefits provided by vectors become more obvious. $\endgroup$ – tommytwoeyes Feb 7 '16 at 1:44
  • $\begingroup$ When you get past mechanics, you'll see more reasons why the use of vector spaces are useful. Just look up Maxwell's equations for a sneak peak! $\endgroup$ – aquirdturtle Feb 7 '16 at 5:26
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What would your "single variable" for direction look like?

You'd need two angles, or something equivalent, to describe it, and two angles and a magnitude is exactly a vector in spherical coordinates.

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