Mathematics of Physics and Engineering

6 Euclidean Geometry

U + V + W

Fig. 1.1.3 Associativity of Vector Addition

Fig. 1.1.4 Multiplication by a Scalar

Multiplication of a vector by a scalar is easily seen to obey the following

algebraic rules:

X(u + v) = Xu + Xv (distributivity over vector addition)

(A + /x)u = Xu + piu (distributivity over real addition)

(AjLt)u = A(/itt) (a mixed associativity of multiplications)

I u = u.

(1.1.2)

In particular, two vectors are parallel if and only if one is a scalar multiple

of the other.

Definition 1.1 A (real) vector space is any abstract set of objects,

called vectors, with operations of vector addition and multiplication by

(real) scalars obeying the seven algebraic rules (1.1.1) and (1.1.2).