**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 Subtraction

Figure 6.7 illustrates the vector difference between and . From the coordinates, we compute .

Note that the difference vector may be drawn from the tip of to the tip of rather than from the origin to the point ; this is a customary practice which emphasizes relationships among vectors, but the translation in the plot has no effect on the mathematical definition or properties of the vector. Subtraction, however, is not commutative.

To ascertain the proper orientation of the difference vector , rewrite its definition as , and then it is clear that the vector should be the sum of vectors and , hence the arrowhead is on the correct endpoint.