- So the main reason you should like polar form is because it makes multiplication and division of complex numbers really, really easy. Here's how we do it. So if you've got two vectors in polar form, z1 and z2, the way that we multiply them together is by taking the moduli of both and multiplying them together and then doing cis and adding the angles together. 

And if we want to divide them, we just do the opposite of that. So we divide the first moduli by the second, and then we subtract the first angle from the second. And it's going to be as simple as that when it comes to multiplying and dividing these complex numbers. If we take a look at this first one, if we want to find z1, z2, we can say that z1, z2 is going to be equal to the moduli multiplied by each other-- 2 times 4 cis-- add the angles together. That's going to be 8 cis 50. That's how easy it is to multiply two vectors in polar form. 

Of course, there is a small trick to it. So let's take a look at this one. So z1, z2 is going to be equal to the moduli multiplied together-- so 3 times 2-- cis the angles added together, pi on 2 plus 5 pi on 6. Now, 3 times 2 is obviously 6. And cis-- so that's going to be 3 pi 6 plus 5 pi on 6-- is going to be 8 pi on 6. 

And then you might say to yourself, ooh, 8 pi on 6, that can be simplified. It sure can. So 5 cis 4 pi on 3. And you might think you're done, but 4 pi on 3 is over here somewhere. And that's outside of the bounds of how we write polar form because we always write polar form between negative pi and positive pi. So we need to rewrite 4 pi on 3 as being cis negative 2 pi on 3. 

All right. So the bones of it, they're still simple stuff. But make sure you're simplifying the angle, and then make sure you're putting it in between negative pi and pi. Those two examples were multiplying. Now we're going to divide, so z1/z2. And we've got these two complex numbers here. But problem-- they're in Cartesian form. So if we're going to divide these, it's going to be easier if we convert them to polar form first. 

Now you already do know how to do that. So I'm going to do it real fast. First up, I found the moduli of z1. There we go. I'm going to do a little bit of trigonometry here, the imaginary component over the real component, pi on 6. But then I need to consider what quadrant this complex number is in. 

So it was in the second quadrant. So pi minus pi on 6 makes 5 pi on 6. And there's our angle. So we can now say that-- we can just finish this off. z1 is equal to 2 cis 5 pi on 6. All right. So there's the first one there. We can also do this second one. 

All right. And so I've gone through the whole lot again, finding the logic, finding the argument. So now we know that z2 is equal to 4 cis pi on 6. From there, it's as simple as using our rule to divide these two now polar-form complex numbers. All right. So z1/z2 is going to be equal to 2/4 cis 5 pi on 6 minus pi on 6. 

And we can obviously simplify that. 2/4 is 1/2 cis-- 5.6 minus pi 6 is 4 pi on 6. Simplifying that-- 2 pI on 3. Now, if this question wasn't so annoying, we'd be finished, but it does say express in Cartesian form. So now we've got to take that lovely polar form, complex number, and convert it back into Cartesian form. 

We expand that cis, and we get 1/2 cos 2 pi 3 plus 1/2 sine 2.3 i. Now, we take that 2 pi on 3, and we move it into the first quadrant. But this is cos in the second quadrant. When we move it into the first quadrant, the sign's going to change. So we get negative 1/2 cos pi on 3 plus 1/2 sine pi on three i. 

Second quadrant, sine is positive. Moving into the first quadrant, that's also going to be positive. From there, we need to know what cos pi on 3 is. We need to know what sine pi on 3 is. All right. So cos pi on 3 is 1/2. Sine pi on 3 is root 3 on 2. We can finish this whole thing off now, and we can say that negative 1/2 times 1/2 is negative 1/4. There's our real component. And 1/2 times root 3 on 2 is going to be root 3 on 4 i. There is our imaginary component. That whole thing is z1 divided by z2. 

All right. The important thing here was the multiplying and dividing of those complex numbers. The next one, when we start raising them to powers, that's just going to be awesome.