**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Vector Addition

Given two vectors in , say and , the

vector sumis defined byelementwiseaddition. 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.

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

triangleby translating the origin of one member of the sum to the tip of the other, as shown in Fig. 6.3.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., = .