Page 267 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Properties of Distance in
If u, v, and w are vectors in and k is any scalar, then:

(a)
(b) if and only if
(c)
(d) (Triangle inequality)
We shall prove part (d) and leave the remaining parts as exercises.
Proof (d) From 2 and part (d) of Theorem 4.1.4, we have

Part (d) of this theorem, which is also called the triangle inequality, generalizes the familiar result from Euclidean geometry that
states that the shortest distance between two points is along a straight line (Figure 4.1.3).

Figure 4.1.3
Formula 1 expresses the norm of a vector in terms of a dot product. The following useful theorem expresses the dot product in
terms of norms.
THEOREM 4.1.6

If u and v are vectors in with the Euclidean inner product, then