Page 161 - Schaum's Outline of Theory and Problems of Advanced Calculus

152 VECTORS [CHAP. 7

If A, B and C are vectors, and m and n are scalars, then
1. A þ B ¼ B þ A Commutative Law for Addition
2. A þðB þ CÞ¼ ðA þ BÞþ C Associative Law for Addition
3. mðnAÞ¼ ðmnÞA ¼ nðmAÞ Associative Law for Multiplication
4. ðm þ nÞA ¼ mA þ nA Distributive Law
5. mðA þ BÞ¼ mA þ mB Distributive Law
Note that in these laws only multiplication of a vector by one or more scalars is defined. On Pages
153 and 154 we define products of vectors.

LINEAR INDEPENDENCE AND LINEAR DEPENDENCE OF A SET OF VECTORS
A set of vectors, A 1 ; A 2 ; .. . ; A p ,is linearly independent means that a 1 A 1 þ a 2 A 2 þ þ a p A p þ þ
a p A p ¼ 0 if and only if a 1 ¼ a 2 ¼ ¼ a p ¼ 0 (i.e., the algebraic sum is zero if and only if all the
coefficients are zero). The set of vectors is linearly dependent when it is not linearly independent.

UNIT VECTORS
Unit vectors are vectors having unit length. If A is any vector with length A > 0, then A=A is a unit
vector, denoted by a, having the same direction as A. Then A ¼ Aa.

RECTANGULAR (ORTHOGONAL) UNIT VECTORS
The rectangular unit vectors i, j, and k are unit vectors having the direction of the positive x, y, and z
axes of a rectangular coordinate system [see Fig. 7-5]. We use right-handed rectangular coordinate
systems unless otherwise specified. Such systems derive their name from the fact that a right-threaded
screw rotated through 908 from Ox to Oy will advance in the positive z direction. In general, three
vectors A, B, and C which have coincident initial points and are not coplanar are said to form a right-
handed system or dextral system if a right-threaded screw rotated through an angle less than 1808 from A
to B will advance in the direction C [see Fig. 7-6 below].

Fig. 7-5 Fig. 7-6