Orthogonality

A vector and one of its components

The unit vectors i and j have a few different properties that make them an excellent basis for two-dimensional vector construction. One such property, which we have mentioned, is that they have been normalized. Another is that they are orthogonal. The mathematical definition of orthogonality is that if the dot product of two vectors equals zero, then the two vectors are orthogonal. Thus, if

v · w = |u| |w| cos(θ) = 0,

then vectors v and w are orthogonal. From this last equation, you can see that orthogonality must occur when cos(θ) = 0, or when θ = 90°. In other words: perpendicular vectors are orthogonal. Why is this a good quality for a basis set to have? Since i and j are perpendicular (or orthogonal), the component of each vector in the direction of the other is zero, and thus neither vector can be made up of any part of the other. And since we’re trying to build vectors from elemental parts, you can hopefully see the sense in requiring that these elemental parts are not at all made up of each other, as that would get kind of confusing.

As a final note, all of this discussion also holds in three dimensions with the unit vector k being added to the mix. All equations and concepts are the same, there is simply one more component involved.

In the signal world, the dot product of two signals must be zero for the signals to be considered orthogonal. See this link.

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Discovery Project II. Vaz, Richard F. 2001. WPI. 14 June 2002

<http://www.ece.wpi.edu/courses/ee2311/temp/dp2.htm>