Complex rotation

- [Voiceover] So now, we've seen rotation by multiplying j by j, over and over again, and we see that that's rotation. Now, let's do it for the general idea of 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 jb. And 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 gonna put ja on this side, ja on this side. And what do we have here? J times j is minus one, so we have minus b plus ja. So, now we have expressions for z and jz. And I wanna plot these on a complex plane and 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'll put b here. And that means that z is at a location in the complex plane, right there. We can plot the dotted lines. That's z in the complex plane. So now, let's put jz on this same plot. Jz 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 jz. And let me draw the hypotenuse of that. This is the vector representing jz, right there. So, now we have a bunch of triangles on the page, and what I wanna 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 sketched in, just imagine in your head that we're gonna 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 rotated 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 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 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. Or 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 jz? And that equals j times r e to the j theta. So, now I'm gonna do a little trick, where I'm gonna represent j in exponential notation. So, if I color in dark here, this is j. The vector j is right there and it has a magnitude of one 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 r e to the j theta. And now the last step is we just combine these two exponents together and we get jz 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 we can rotate any complex number by 90 degrees if we multiply it by j. We're gonna get to apply this kind of transformation to working out the current and voltage relationships in inductors and capacitors, and that'll happen in a couple videos from now.