 In my last video we discussed what the quadrant system was. Now I'm going to get down to brass tax and show you why it's important. Now the reason why we use quadrants is because of vectors. And a vector is something that contains magnitude, so how large it is, and direction. Which way is it heading? A vector can show up in any of the four quadrants, one, two, three, or four. In this case this vector here is whatever size this is at whatever angle which is the direction in quadrant one. There's two ways that we can look at vectors. We can look at them in what's called polar form or rectangular form. In this video I'm going to go through about converting from a polar vector to rectangular coordinates. So let's take a look here. Now in this example here I'm saying that this vector is twenty volts and it's heading off at thirty-five degrees. The way that we've expressed this vector is in what is known as polar form. So it's giving you its magnitude and at what angle it's going at, so magnitude and direction. Now when we want to add vectors however it would be a lot easier for us to kind of break these guys down into an X and a Y coordinate. And we can do that now because you guys have been given the power of trigonometry. We know that this guy here is twenty volts, so that's my hypotenuse. If I want to figure out what my X is I can figure out this guy here and this guy here. That makes us a right triangle so I can use trigonometry. Now if I want to figure out what the bottom is here, the adjacent to this angle, I'm going to use cos. When I want to figure out what my Y is I'm going to use sine because I've got cos is adjacent over hypotenuse and sine is opposite over hypotenuse. So let's kind of break this down here. Using cos I can figure out what my X coordinates are. I can take 20 volts times the cos of 35 degrees and I get that my X is going to work out to be 16.38 or 16.4 if you want to get funky with the rounding. Now I'm going to take the same thing, I'm going to go 20 times the sine of 35 degrees, it's just a little shortcut instead of having to transpose all the time. And I'm going to get my Y coordinate. Actually I've got that 16.4 there. Now my Y is 20 times sine 35 is 11.47 which is going to be this side right here. So I've got that punched in there so basically now I have my X and my Y which is the same thing as this here. So this form 20 volts at 35 degrees is your polar form. Your X being 16.4 and your Y being 11.5 that is what's known as rectangular. And we differentiate our Y from our X by throwing a little J in front of it. And you can go to my notes in my lesson plans that I'll have and it will explain, give you a little explanation as to why we use J. So there we go, we've got 16.4 or 16.38 and J 11.47 that is the rectangular form. This guy and this guy they are identical. So that's in quadrant one. Now in quadrant one we remember that it's both positive X and positive Y. What happens when we end up in quadrant number two? Boom here we got 60 volts heading at 150 degrees. So we know that in quadrant one was between 0 and 90, quadrant two is between 90 and 180. So that's going to throw this vector into quadrant number two. Now the same thing is going to go here. All we have to do is go 60 times the cos of 150 is going to give me my X and 60 times the sine of 150 is going to give me my Y. 60 times the cos of 30 degrees gives me 51.96 for my X size. Now let's take a quick look at this before I go on. I've got negative 51.96 that's negative because I've got my point of origin here and on the X it's negative remember from my last video. So that's very important if you forget that you're going to screw yourself over when it comes to adding vectors do not forget the polarity. Now if I take 60 times the sine of 30 degrees I end up with my Y being 30. Now it is above the X axis right I'm still in this positive sense here. So I could say that this guy is going to be positive 30. So now I can say that my coordinates, my rectangular expression of this vector is going to be negative 51.96 and J30. So that's giving us our second quadrant. Now why don't we fool around we've got two more quadrants to go through why don't we just do them and get them over with and that way you guys can see how it works for every single quadrant. Now in quadrant number three I've got 100 volts which is the size of magnitude and 200 degrees which definitely throws me in the third quadrant because anything between 180 and 270 gives me quadrant number three. Again I'm going to go 100 times the cos of 200 degrees gives me my X which is going to be negative and in this case my Y is also going to be negative. So 100 times the cos of 20 gives me negative 93.97. See I've got that negative in front of it. 100 times the sine of 20 gives me 34.2 but I put down negative 34.2 down there and then I can express this as negative 93.97 and negative J or J negative 34.2. So again this is polar this is rectangular and let's finish it up here in quadrant number four. I've got 90 volts at 320 degrees. So again same thing 90 times the cos of 320 degrees is going to give me a positive X and then my Y however is going to be negative. So again I cannot stress enough the importance of those polarities. So 90 times the cos of 40 degrees gives me 68.94 right here so that's going to be a positive 68.94 and next up I've got to go 90 times the sine of 320 degrees which would give me a negative number. And I've got that as 57.85 which is right there so that will be a negative 57.85. And there we have it. If I've got my polar form which is 90 volts at 320 degrees and these two guys here can be expressed as 68.94 negative J 57.85. And that's how you convert rectangular or sorry polar to rectangular. Polar is when you've got the magnitude and the direction so you've got your angle and the size and rectangular is your X and your Y coordinates and remember that our Y is always preceded with a J just so we can tell the difference between the two and also again I know I keep saying it but do not forget your polarity.