Concept guide
Euclidean length: computing the distance between two points exactly from coordinates
In the Euclidean plane represented by ℝ², let P=(1,−2), Q=(4,2), R=(2,3). Compute d(P,Q)=5 exactly; compute d(P,R)=√26 and prove by comparing squares that 5.09<√26<5.1, showing that 5.1 is only an approximation. For general real coordinates, also prove that the distance is defined and nonnegative, that d(Q,P)=d(P,Q), that d(P,P)=0, and that d(P,Q)=0 ⇔ P=Q.