I'm learning multivariable calculus right now and I have a brief linear algebra background. I'm very curious why we have "vector-valued" functions rather than ones that output a point in n-dimensional Cartesian space. I understand why vectors have physical application, but when parameterizing with a vector-valued function, it's the output is really (from my understanding) what the tip of the vector traces out. So in summary, I'm curious why we treat the outputs as vectors rather than just points like we would do with a scalar- valued function.

  • $\begingroup$ Given a point, you can find a vector from origin to that point; and given a vector, you can find a point that is pointed to by the vector. So the two notations are the same. $\endgroup$ – peterwhy Jun 18 '17 at 16:08
  • $\begingroup$ But sometimes you may have to add or subtract results of functions, e.g. when taking derivative, and adding point does not have an obvious meaning, but adding vectors does. $\endgroup$ – peterwhy Jun 18 '17 at 16:11

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