Page 199 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

3.2 In this section we shall establish the basic rules of vector arithmetic.

NORM OF A VECTOR;
VECTOR ARITHMETIC

Properties of Vector Operations

The following theorem lists the most important properties of vectors in 2-space and 3-space.
THEOREM 3.2.1

Properties of Vector Arithmetic
If u, v, and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold.

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)

Before discussing the proof, we note that we have developed two approaches to vectors: geometric, in which vectors are
represented by arrows or directed line segments, and analytic, in which vectors are represented by pairs or triples of numbers
called components. As a consequence, the equations in Theorem 1 can be proved either geometrically or analytically. To
illustrate, we shall prove part (b) both ways. The remaining proofs are left as exercises.

Proof of part (b) (analytic) We shall give the proof for vectors in 3-space; the proof for 2-space is similar. If ,