Theorem. Let V be a Euclidean vector space then
the distance function has the following properties:
- d(A,B)> or equals 0, d(A,B)=0 if and only if A=B.
- d(A,B)=d(B,A).
- d(A,B)< or equals d(A,C)+d(C,B) (the triangle inequality).
The proof of 1 and 2 is left as an exercise. Let us prove 3.
We have:
d(A,B)=||A-B||=||A-C+C-B||< =||A-C||+||C-B|| (the triangle inequality for
norms)=d(A,C)+d(C,B).
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