If F =A−B denotes the difference of two vectors A and B, then F is determined
by the above rule for vector addition by writing F =A+ (−B). Thus, subtraction
of the vector B from the vector A is represented by the addition of the vector −B
to A. In figure 6-2 observe that the vectors A and B are free vectors and have
been translated to appropriate positions to illustrate the concepts of addition and
subtraction. The sum of two or more force vectors is sometimes referred to as the
resultant force. In general, the resultant force acting on an object is calculated by
using a vector addition of all the forces acting on the object.
Vectors constitute a group under the operation of addition. That is, the following
four properties are satisfied.
1. Closure property If A and B belong to a set of vectors, then their sum A+B
must also belong to the same set.
2. Associative property The insertion of parentheses or grouping of terms in vector
summation is immaterial. That is,
(A+B) + C =A+ (B+C) (6 .50)
3. Identity element The zero or null vector when added to a vector does not produce
a new vector. In symbols, A+0 = A. The null vector is called the identity element
under addition.
4. Inverse element If to each vector A, there is associated a vector E such that
under addition these two vectors produce the identity element, and A+E = 0 ,
then the vector E is called the inverse of Aunder vector addition and is denoted
by E=−A.
Additional properties satisfied by vectors include
5. Commutative law If in addition all vectors of the group satisfy A+B =B+A, then
the set of vectors is said to form a commutative group under vector addition.
6. Distributive law The distributive law with respect to scalar multiplication is
m(A+B) = mA+mB, where mis a scalar. (6 .51)
Definition (Linear combination)
If there exists constants c 1 , c 2 ,... , c n, not all zero, together with a set of vectors
A 1 ,A 2 ,... , An, such that
A =c 1 A 1 +c 2 A 2 +··· +cnAn,
then the vector A is said to be a linear combination of the vectors A 1 ,A 2 ,... ,An.
