The distance between two points in different dimensions

Question

Suppose I want to know the difference between point A = (0,0,1), which is in ℝ³, and point B = (1,0), which is in ℝ². Does it make sense to measure the distance between these two points?

Answer 1

Here you are searching for a metric on $X = \mathbb R^3 \sqcup \mathbb R^2$

However, the natural metric induced on metric spaces by taking a disjoint union is to put the distance between different dimensioned objects to 1, and to use the distance functions on the projection if the dimensions are equal. Sure you may use another function as the distance between two different dimensioned vectors, but that would be making a choice -- by setting it to 1 you have kept it natural.

This question here shows why that does give you a metric on disjoint union: Disjoint Union of Metric Spaces is a Metric Space

Continuing however, there are obviously ways that different dimensioned objects could be related by some algebra, statistics, or other reasons, like could be used to encode some graph structure and then the other vector is various computations of the first vector. But you would need to understand the context of why you are getting elements from $X$ specifically, and within that context, you might be able to define your own metric function which will let you compute this distance in a way which is meaningful to your situation involving $X$.

Answer 2

Not really, the Euclidean distance is defined on a certain vector space, a certain $\mathbb{R}^n$, in your case. So it would only make sense to measure distances between two points of the same vector space.

That being said, $B$ can be inside a certain plane embedded in $\mathbb{R}^3$, for example the $\{z=0\}$ plane, in which case the 3D coordinates of $B$ would be $(1,0,\bf{0})$, and so the distance $AB$ is properly defined.