# What do vector size and direction really mean in vectors that are not position vectors?

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?

• @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. – Arthur Oct 20 '20 at 17:21
• @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. – Arthur Oct 20 '20 at 17:48
• (Or rather $5^2+3^3$.) – Arthur Oct 20 '20 at 17:54