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Vector Addition

Given two vectors in $ \mathbb{R}^N$ , say

\underline{x}&\isdef & (x_0,x_1,\ldots,x_{N-1})\\
\underline{y}&\isdef & (y_0,y_1,\ldots,y_{N-1}),

the vector sum is defined by elementwise addition. If we denote the sum by $ \underline{w}\isdef \underline{x}+\underline{y}$ , then we have $ \underline{w}_n = x_n+y_n$ for $ n=0,1,2,\ldots,N-1$ . We could also write $ \underline{w}(n) = x(n)+y(n)$ for $ n=0,1,2,\ldots,N-1$ 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 $ N=2$ , $ \underline{x}=(2,3)$ , and $ \underline{y}=(4,1)$ .

Figure 5.2: Geometric interpretation of a length 2 vector sum.

Also shown are the lighter construction lines which complete the parallelogram started by $ \underline{x}$ and $ \underline{y}$ , indicating where the endpoint of the sum $ \underline{x}+\underline{y}$ lies. Since it is a parallelogram, the two construction lines are congruent to the vectors $ \underline{x}$ and $ \underline{y}$ . 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.

Figure 5.3: Vector sum, translating one vector to the tip of the other.

In the figure, $ \underline{x}$ was translated to the tip of $ \underline{y}$ . This depicts $ y+x$ , since ``$ x$ picks up where $ y$ leaves off.'' It is equally valid to translate $ \underline{y}$ to the tip of $ \underline{x}$ , because vector addition is commutative, i.e., $ \underline{x}+\underline{y}$ = $ \underline{y}+\underline{x}$ .

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``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
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University