 In this video, we're going to be going over how to convert from rectangular form, which is when we break it down to an x and a y coordinate, when we take a vector and we work out what its x is and its y, we're going to take the rectangular form and convert it to polar. So what we're going to do here is we're going to start out with this vector right here. We've got an x of 25 and a j of 50, which means that we've got an x of 25 and a j of 50. Here means we're going to be in quadrant number one. So let's throw some numbers at that, throw a picture up here just so we get our head wrapped around it. So I've got 25 on my x and I've got 50 on my y. So let's just draw the vector in. So I've got 25 here and 50 here. So now what we need to do is figure out what our hypotenuse is going to be and this is our resultant, which is the first part. This is the magnitude that we're looking for when we're talking about the polar form as we learned in our last video. So we're going to go 25 squared plus 50 squared gives me the square root of. We end up with a resultant of 60. So that is the magnitude. Now we need to figure out what direction it's heading in and that would be our angle. So what I'm going to do is use cos. I'm going to go 25 divided by 60 and then I'm going to inverse cos that. And if you wanted to know why I'm doing that, you just need to watch the videos on trigonometry, but I'm taking my adjacent side and my hypotenuse and adjacent over hypotenuse is the ratio of cosine. So I do that, the cos of 25 divided by 60 inverse cos that I get 65 degrees. So now I've done it. I've figured out that my magnitude is 60. That's my resultant, the hypotenuse. The angle here is 65 degrees because I'm in quadrant one, which is this quadrant here. Again, watch the video in quadrants. It's anywhere between 0 and 90 degrees. I can just easily say that it is 60 at 65 degrees. So there you go. 60 at 65 degrees is the polar form of 25J50. That's in quadrant number one. Let's move around the four quadrants like we've done in the previous videos and let's see what happens. So in this example now, we've got negative 45, which is negative 45s on my x, so it means it's heading this way. 60 means it's heading up. So that puts me in quadrant number two. So let's get that started to be drawn up. So I've got negative 45 right here and I've got 60 here. Then through the power of Pythagoras, 45 squared plus 60 squared gives me the square root of 75. 75, again, is my resultant. Now what we're going to do is we're going to figure out what the angle is in here. And this part's really important. So let's take 45 divided by 75 and inverse coset that. And that gives me 53.1 degrees. So now I've got all the sides and the angle here. Now, if I was going with polar form, I could say 75, but I cannot say at 53.1 degrees because we need to look at which quadrant we are in. We're in quadrant number two, which means that our angle has to be anywhere between 90 and 180 degrees. Well, we know that this guy here is 53.1 degrees and we're looking at the angle as in reference to zero. So we want this angle. So what we're going to do is just take this guy here, 180, minus the 53.1 degrees here to get my actual angle in this quadrant. And I get 129.6 degrees. So now I can easily say that this guy here, this negative 45j 160, it's got a polar form of 75 at an angle of 126.9 degrees. And there you go. That 126.9 tells me we are in quadrant two as well. Let's move it around. Let's go on to quadrant three. So now I have negative 120, negative j 40. So negative 120, negative j 40. We know for sure that we are in quadrant number three. Let's start playing around with this. Get it drawn up here, 120, so that's the negative 120 puts me on this side of the point of origin. 40 is down here. So now I need to figure out what my resultant is or my hypotenuse. And that works out to be a hypotenuse of 126.5, which is my resultant. So again, we're getting there. Now we just need to figure out what this angle is. I'm using cos, I'll go 120 divided by 126.5 and inverse cos, that guy, and I get an angle of 18.5 degrees. Now again, we are in the third quadrant. I can't go ahead and say it's 126.5 at 18.5 degrees, because then you think I'm talking with the first quadrant. So before, in the last one, we went 180 minus that angle that we worked out to get the angle. Now we're going to go 180 plus the 18.5, and that will give me my angle, which is 198.5. So now we can say that our rectangular form, which is negative 120, negative J40, has got a polar form of 126.5 at an angle of 198.5 degrees. There we go. So we've done the first quadrant. We've done the second quadrant. We've done the third quadrant. Let's move on to the fourth quadrant. Triangle 30. So we're moving this side of the point of origin, negative J75 going below there. Again, we are in quadrant number four. So as we've done before in every single quadrant, all we have to do is draw that in there, get your triangle drawn in, 30, 75. Using Pythagoras, we figure out what my resultant is, 80.7. Now we're going to work out what our angle is here, 68.2 degrees. So now we can say that we've got a resultant of 80.7, and we know that this little angle in here is 68.2 degrees. Now we said that this angle on this side was 180 minus, the angle on this side is 180 plus. We could say that the angle over here is 360 minus, and that would give us an angle of 291.8 degrees. So we could say that this guy, 30, and negative J75 rectangular is the same as saying 80.7 at 291.8 degrees. But what we can also say when we're dealing with the fourth quadrant is we can work our way backwards. Now, I won't go too much into why we can do this. You can do this with every quadrant, but we'll do this with the fourth quadrant because it's somewhere between 0 and 90 backwards, so it'd be 0 and negative 90 this way. We can look at it and say, okay, we've got 80.7 at an angle of negative 68.2 degrees. It's the same thing as long as you work backwards. Now we generally only do this in the fourth quadrant. So you can work out what this degree, this angle is, and throw a negative in front of it using this resultant, and you can work out what that is. Now it's very important that we understand how to do these conversions because in our next video we're going to start talking about how to add vectors. See you there.