Given two vectors in , say
the vector sum is defined by elementwise addition. If we denote the sum by , then we have for . We could also write for if preferred.
The vector diagram for the sum of two vectors can be found using the parallelogram rule, as shown in Fig.5.2 for , , and .
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.5.3.
In the figure, was translated to the tip of . This depicts , since `` picks up where leaves off.'' It is equally valid to translate to the tip of , because vector addition is commutative, i.e., = .
Signals as Vectors