If we take a Cartesian system with length over length then the vector magnitude has length as unit of measurement. I get that. Let's say we have cups of coffee as the x axis and cost in dollars as the y axis. If we take a point (1cup, 1dollar) and we create the point vector $u=\{1\ cup, 1\ dollar\}$ then we can calculate the vector length which is: $|u|=\sqrt{1^2 \ cup + 1^2\ dollar}$.
1--what is the result?
2--what unit of measurement will it have?
3--how can we interpret the vector length with that unit of measurement? If the result is $\sqrt{2} * \sqrt{cup^2 + dollar^2}$ how can we interpret this?
Thank you
Short answer: if the $x$ axis measures cups and the $y$ axis measures dollars then the length of a vector makes no sense. There is no reasonable way to interpret it. Try to imagine what would happen if you converted dollars to euros.
In this problem you don't really have a Euclidean vector space. If you did, you could rotate the coordinate system. That's clearly nonsense here.
There are constructions that do make sense here that don't in a two dimensional vector space. For example, you might care a lot about the value of $y$ divided by the value of $x$: the cost of coffee in units dollars per cup.
Even when the units on the axes are the same the length makes sense only if the axes are measuring the same thing. You can't think about the Euclidean vector length if you're plotting the cost of a your restaurant food versus the amount of your tip even though both are measured in dollars. Then you might want the taxicab metric: sum the (positive) coordinates to get the cost of a meal.