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I know that vectors are objects that have both magnitude and direction. Let's say we have a vector r that represents a house as an example, and it is r = [a, b, c] (a=number of rooms, b=number of floors, c=price). For this vector the idea of finding its magnitude does not make any sense to me, what would it mean? Also what would the direction or the angle between two house vectors mean?

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Not all vector spaces have a sensible dot product, and thus not all vector spaces have a sensible notion of length and direction (other than two vectors being or not being parallel, and one vector being longer or shorter than another, parallel vector).

Your vector space is a case of this, where the non-existence of a sensible dot product is a result of the application, rather than a mathematical inability to impose any dot product on the space. (A vector space where we mathematically cannot impose a dot product implies the negation of the axiom of choice. So that's not an easy thing.)

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  • $\begingroup$ Okay! So, for the same vector in my question, we can perform dot prodcts and calculate its magnitude mathematically even if it is not sensible. Another thing that I am wondering about is that each element of the vector is representing different quantities like price and number of rooms. Is it mathematically correct to use pythagorean theorem to calculate the magnitude because they are different like oranges and bananas? $\endgroup$ – Salvator Oct 20 '20 at 17:17
  • $\begingroup$ @user88731 Any dot product you impose will imply some form of the Pythagorean theorem on your space (although it might not be that the coordinate axes are orthogonal with the given dot product, so it's not necessarily as simple as $a^2+b^2+c^2$). So if you have a dot product, you have Pythagoras. $\endgroup$ – Arthur Oct 20 '20 at 17:21
  • $\begingroup$ What I mean is that a^2 + b^2 + c^2 have 3 different values each representing 3 different things. As I said it is kind of like adding bananas and oranges together. $\endgroup$ – Salvator Oct 20 '20 at 17:28
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    $\begingroup$ @MustafaShujaie No, it is the basis vectors that have units. The coefficients are unitless. So as far as pure arithmetic goes, there is no issue with $a^2+b^2+c^2$. You have $5$ oranges and $3$ apples, and the Pythagorean formula is like telling you to calculate $5+3$. Makes no practical sense, but entirely doable. $\endgroup$ – Arthur Oct 20 '20 at 17:48
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    $\begingroup$ (Or rather $5^2+3^3$.) $\endgroup$ – Arthur Oct 20 '20 at 17:54
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There doesn't need to be a physical interpretation for your examples. Vectors, in full generality, don't need to have a magnitude and direction. In your case, the vector's magnitude and direction are just the magnitude and direction you get when you draw it in 3D space---no physical meaning.

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