4 1 Introduction
Fig. 1.1 Vector addition
a+b=b+a. (1.1)As a side remark, one may ask: how was the rule for the vector addition conceived?
A vectoracan be associated with the displacement or shift along a straight line of
an object, from point 0 to pointA. This is the origin for the wordvector:itcarries
an object over a straight and directed distance. The vectorbcorresponds to a shift
from 0 to pointB. The vector additiona+bmeans: make first the shift from 0 to
pointAand then the additional shift corresponding to vectorb, which has to start
from pointA. For this reason, the tail of the second vector is attached to the head of
the first vector in the vector addition operation.
When three vectorsa,bandcare added, it makes no difference when first the
vector sum ofaandbis computed and then the vectorcis added or whenais added
to the sum ofbandc:
(a+b)+c=a+(b+c). (1.2)
The vector sum is also used to define the difference between two vectors according to
a+x=b → x=b−a. (1.3)When the vectorbin (1.3) is equal to zero, then one has
x=−a. (1.4)This vector has the same length asabut the opposite direction, i.e. arrowhead and
tail are exchanged.
For real numberskand, the following rules hold true for any vectora:
(k+)a=ka+a,
k(a)=(k)a, (1.5)
1 a=a.