Consider two non-zero complex numbers *z*_{1} and *z*_{2}. If the arguments of these two numbers differ by π/2 (the complex numbers, when treated as vectors in the complex plane, define perpendicular vectors), what does this say about the value of ?

First, we note that , and that the argument of the product of two complex numbers is the sum of the arguments of the numbers. Thus,

Now, we said that the two arguments differ by π/2; thus

but that is the argument of a purely imaginary number; so

.

We see that this argument is reversible; if , then

,

and the vectors representing two nonzero complex numbers *z*_{1} and *z*_{2} are perpendicular if and only if .

Note that if we give and , then

.

Note that expressed as conventional vectors and , then , so

is equivalent to , confirming perpendicularity of the vectors.

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Tags: Complex Numbers, Math, Monday Math

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