The definition of “vector” in math and physics

  • #1
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I'm learning APL and this is how a vector is defined https://tryapl.org:

All data resides in arrays. An array is a rectangular collection of numbers, characters and arrays, arranged along zero or more axes. We can use more specific terms for some arrays, like a single number is a scalar, a list is a vector, and 2D arrays are matrices. Vectors can be formed by just placing elements next to each other:​

I see that in math a list of numbers is called a vector.

In physics a vector is a directed magnitude. So vector in physics is a set of only two numbers. One of these numbers is interpreted as magnitude and the other is interpreted as direction.

My question is: Is the above description correct?
 

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  • #2
Orodruin
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So vector in physics is a set of only two numbers. One of these numbers is interpreted as magnitude and the other is interpreted as direction.
This is not a correct conclusion. Direction cannot be described by a single number (unless you are just looking at two dimensions).
 
  • #3
jbriggs444
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Vectors are more formally treated in a discipline known as "linear algebra". It turns out that the ability to treat a vector as a one dimensional array of numbers can then be traced to being able to identify a "basis" for a vector space. Individual vectors within the space can then be identified with unique linear combinations of basis vectors and the linear combination can [sometimes] be described as a one dimensional array of numbers.

https://en.wikipedia.org/wiki/Vector_space
 
  • #4
Stephen Tashi
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In physics a vector is a directed magnitude.
At various stages of our education, we learn different definitions for the same word. Different definitions for the same word are used in different fields of study. So you can expect to hear different definitions of the word "vector".

As @jbriggs444 indicates, from an advanced point of view, a vector is an element of a mathematical structure that satisfies certain axioms. An ordered list of numbers is one example of a vector but an ordered list of numbers is not the only example of a vector. When you say "A vector is... such-and-such", you must think about whether your are saying a vector is identical to something or whether it is merely one example of something.

Not all physical quantities that can be treated as vectors have a "magnitude and direction". Those that do have a magnitude and direction can usually be regarded as vectors.
 
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