Multiplying Polar Coordinates
What do we obtain when we multiply the polar coordinates:
( r 1 , θ 1 ) × ( r 2 , θ 2 ) ?
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1 solution
Yeah I was wondering if we still treated them like vectors or if they had their own outer product?
This is really the product of complex numbers in polar form, which returns "another complex number" also in polar form.
When we multiply two complex numbers their absolute values (moduli) get multiplied and their arguments are added. Therefore ( r 1 , θ 1 ) × ( r 2 , θ 2 ) = ( r 1 × r 2 , θ 1 + θ 2 ) .
A vector dot product yields a scalar, while a vector cross product yields a vector which is normal to the original two.
It's interesting how complex numbers differs from vectors in that way, even though complex numbers are often treated as if they were vectors.
What does it mean, exactly, to "multiply polar coordinates", if not talking about a product of complex numbers? This one has got me wondering about other possible spaces where this can make sense.