## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Vector Addition

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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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.

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.

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., = .

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