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# Complex rotation

## Video transcript

so now we've seen rotation by multiplying J by J over and over again and we see that that's that's rotation now let's do it for the general idea of any any complex number so if I have a complex number we'll call it Z and we'll say it's made of two parts a real part called a and an imaginary part called B so now what I want to do is what happens if we multiply Z by J one time J times Z and that equals J times a plus J B let's just multiply it through equals J times a plus J times J times B a and B have now switched places so we're going to put J a on this side J a on this side and what do we have here J times J is minus 1 so we have minus B plus J a so now we have expressions for Z and J Z and I want to go a plot these on a complex plane and see what see what they look like and here's the real axis here's the imaginary axis and let's first plot let's plot Z let's say Z has a large real value and that would be a and let's say that B is a smaller value we put B here and that means that Z is at a location in the complex plane right there if we can plot the dotted lines that's Z in the complex plane so now let's put J Z on the same plot J Z has it has a real component of minus B so that would be right about here here's minus B and it has a imaginary component of plus a so let's swing a a goes all the way up to about here and so that's the location of J Z and let me draw the hypotenuse of that this is the vector representing J Z right there so now we have a bunch of triangles on the page and what I want to demonstrate is that this angle right here is 90 degrees so one way to do that let's see if we can do that let's say this angle here is Theta that's the angle right there now this triangle here this triangle that we've sketched in just imagine in your head that we're going to rotate that angle up until the a leg of that triangle is resting right here on the imaginary axis so this triangle rotates up to become this triangle here since we moved that triangle we know that this angle here that's also theta it's the same triangle just rotate it up and what does that make this angle here this angle here that equals 90 degrees minus theta so if I combine this theta angle with this angle here what do I get theta plus 90 degrees minus theta and the we get 90 degrees so we just showed that this angle right here is a 90 degree angle that demonstrates that any complex number Z if I multiply it by J that results in a positive rotation of 90 degrees so let's do this rotation again only this time instead of using the the rectangular coordinate system let's use the exponential representation so in the exponential notation we say in general Z equals some radius times e to the J theta where this is the angle theta and R is the length of this hypotenuse here to get out to Z so what is in this notation what is J Z and that equals J times our e to the J theta so now I'm going to do a little trick where I'm going to represent J in exponential notation so if I color in dark here this is this is J the vector J is right there and it has a magnitude of 1 and it points straight up on the imaginary axis so I can represent J like this I can say J is e to the J 90 degrees that's equivalent to this J here and it's multiplied by our e to the J theta and now the last step is we just combine these two exponents together and we get J Z equals R times e to the J theta plus 90 degrees so in exponential notation we get this vector here we go an additional 90 degree rotation and we go out the same distance we had originally R so now we've shown that we can rotate any complex number by 90 degrees if we multiply it by J we're going to get to apply this kind of transformation to working out the current and voltage relationships in inductors and capacitors and that that will happen in a couple of videos from now