Euclidean Space

Metric Space:

Any space with a distance metric d(P,Q) defined on its elements.

Distance Metric:

• Metric d(P,Q) must satisfy the following axioms:
  1. 
  2. d(P,Q) = 0 iff P = Q.
  3. d(P,Q) = d(Q,P).
  4. .
• Distance is intrinsic to the space, and not a property of the frame.

Euclidean Space:

Metric is based on a dot (inner) product:

Note:

• Dot product is defined on vectors.
• Distance metric is defined on points.

Dot product:

Norm:

Angles:

Perpendicularity:

Perpendicularity is not an affine concept!