Given two vectors in , say
and , the vector sum is defined byelementwise addition. If we denote the sum by ,
then we have for .
The vector diagram for the sum of two vectors can be found using the
parallelogram rule, as shown in Fig. 6.2 for , , and.
Figure 6.2:Geometric interpretation of
a length 2 vector sum.
Also shown are the lighter construction lines which complete the
parallelogram started by and , indicating where the endpoint of the
sum lies. Since it is a parallelogram, the two construction lines
are congruent to the vectors and . As a result, the vector sum is
often expressed as a triangle by translating the origin of one member
of the sum to the tip of the other, as shown in Fig. 6.3.
Figure 6.3:Vector sum with translation of
one vector to the tip of the other.
In the figure, was translated to the tip of . It is equally valid
to translate to the tip of , because vector addition is commutative, i.e., = .
, say
and
, the vector sum is defined byelementwise addition. If we denote the sum by
, then we have
for
.
,
, and
.
and
, indicating where the endpoint of the sum
lies. Since it is a parallelogram, the two construction lines are congruent to the vectors
.
