Page 200 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

, and , then

Proof of part (b) (geometric) Let u, v, and w be represented by , , and as shown in Figure 3.2.1. Then

Also,
Therefore,

Figure 3.2.1
The vectors and are equal.

Remark In light of part (b) of this theorem, the symbol is unambiguous since the same sum is obtained no matter

where parentheses are inserted. Moreover, if the vectors u, v, and w are placed “tip to tail,” then the sum is the vector

from the initial point of u to the terminal point of w (Figure 3.2.1).

Norm of a Vector

The length of a vector u is often called the norm of u and is denoted by . It follows from the Theorem of Pythagoras that the

norm of a vector in 2-space is

(1)

(Figure 3.2.2a). Let be a vector in 3-space. Using Figure 3.2.2b and two applications of the Theorem of
Pythagoras, we obtain

Thus

(2)