 Let's talk about area as a vector. We'll see how to calculate its direction. We'll take some examples and we'll also talk about why we do that. So let's start there. Let's start at the why. I always wondered, when I first learned about area as a vector, I always wondered, why do people do that? Why do they take something as so nice as area and convert it into vector? Why do you even wanna do that? And the answer is it actually helps us in physics. Let me give you an example. We've already seen in a previous video how to calculate flux. If you have an electric field A, sorry, E going through some area A, then the flux is just the product of the electric field multiplied by the perpendicular component of that area. And we'll talk all about that in the previous video. Now, what I want to focus on is when you wanna try and communicate this with someone, how are you gonna say that? You're gonna say it's the electric field multiplied by what? Area perpendicular. What is this area perpendicular? It's the perpendicular component of this area perpendicular to the electric field. So many words, right? So many words you have to use to communicate that. The whole idea behind making area as a vector or at least one of them is to make this communication easier, simpler. That's the whole idea behind maths, right? That's why we use equations, symbols to make as less words as possible, right? And making area a vector will actually help us. Now, before we do that, you might say, wait a second, you can actually just call this angle as theta. And then you could say that flux phi is just e times, a perpendicular is basically a times cos theta, trigonometry. And again, we've done this in detail in the previous video. So for more details, you can always go back. So you can just do this, right? Isn't this easier? Yeah, but now I ask you, what is angle theta? And how do you explain that? Well, again, you have to say angle theta is the angle between the area and the component of the area that's perpendicular to the electric field. Again, so many words you have to use, right? So that's how do we make it efficient by making area as a vector. You'll see how, all right? So let's come back over here and let's see how to treat area as a vector. So if you take this example, if you have an area like this nice and flat rectangular area, the way to think of it as a vector is you just draw an arrow mark perpendicular to this area. So in this case, if I draw an arrow mark perpendicular to the area this way, that is the direction of our area vector. So the direction is always perpendicular and the size will be proportional to the size of this area. So if this area was bigger, I would have drawn it bigger. So in this case, we will say our area is pointed in the positive z direction. You got it? Draw perpendicular to the area and that's the direction. Let's look at this one. Can you try this one? You have a circle which is in the zx plane. In what direction would the area be if you were to treat it as a vector? Can you pause and try? All right? You have to draw perpendicular. So the perpendicular over here would point this way and therefore we could now say, hey, the area points, so the area, sorry, the area direction is in the positive y direction. And I can give you the other way around. I can give you, I can now give you the area vector. Let's say there's an area vector which points in the x direction. This is the area vector. Can you imagine where that area would be, how that area would look like? Yeah, again, it's gonna be perpendicular. So our area is gonna be somewhere in the zy plane. So you get that, right? So to draw, so think of area as a vector, just draw a perpendicular to it. That's the direction for it. Done, that's it. Now, if you're really curious, you might say, but wait a second, I could have drawn two perpendiculars. I could have drawn one up, but I could have also drawn one down. So how do I know whether the direction is in the positive z direction or the negative z direction? And the same thing I would have done over here. In this, I could have drawn another area vector. I could have drawn the perpendicular this way. So how do I know whether the direction is in the positive y direction or in the negative y direction? How do you know that? And the answer is, it can be arbitrary over here. So can we set a standard? No, we can't set a standard. It's a little hard to set a standard because up and down, I mean, can you set a standard saying that, let's take upwards as positive. You can't because up and down are arbitrary. Left and right are arbitrary. So when you're dealing with area, so that's one problem is we don't know which direction should we consider. In most cases, it will be arbitrary. We can't set a standard, but we don't have to worry about it because most of the times when you're dealing with this area vector, in physics it's gonna be, say, in this unit, it's gonna be somewhere when you're talking about flux. Most of the time, you'll be dealing with a closed surface, not an open surface like this. Let me give you an example of what I mean. You will be dealing with, say, a surface which is like closed in three dimensions. It'll be closed. And over here, when you are dealing with an area vector, so let's say you take a tiny, let me draw here, let's say you take this tiny piece and you wanna treat this as an area, sorry, treat this as a vector. And over here, again, you can draw two arrow marks. You can draw one that's pointing out of this particular object and you can draw one more that's pointing in to the particular object. Now outward and inward, that can be standardized. And what we like to do is we like to say, choose outwards as the standard. So long story short, when you're dealing with a closed surface, we do have a standard. We will only choose the outward direction as the direction of the vector and we'll ignore the inward direction. Let me take another example that you can try. Let's say we take a cylinder which is a closed object. Now imagine I consider this area down here, circular area down here. Can you pause and tell me what will be the direction of that vector? Can you pause and tell? All right, let's do this. So we can choose two directions. We can choose, you should always choose perpendicular to it. So you can choose upwards perpendicular or you can choose, so let me do it in a different layer. Okay, you can choose upwards perpendicular or you can choose downwards perpendicular, but outwards is our standard. So this is what we'll choose. So we'll say over here, the direction is in the negative Z direction or we'll just say downwards and this is wrong, we don't choose this. What about for the, this one, what about for this one? Well again, we can choose to, we can choose out and you can choose inward, but we'll always choose outwards as our standard. So this is going to be the right direction. So notice, even though these two surfaces are pretty much identical, they have the same area, they are pretty much the same orientation. Vectorially, they're not the same. They're in the opposite direction because we always choose the vectors in the outward direction. Okay, so for closed surfaces, there's no problem. Our standard is outwards. For open surfaces, you can't say that because there is no outward, there is no inward. But there's also still, even for open surfaces, there is a way to standardize the direction and we'll not talk about it. It's not really important for this particular view. There is a way to do that. We'll talk about it in some future videos. But as of now, don't worry about them. For closed surfaces, outward is our standard direction. And now I'm sure you are waiting to see how this helps us in communicating this much more efficiently. Let's see, if I try to treat this area now as a vector, let me zoom in over here. So if I treat this area as a vector and let me draw the vector for this area in this direction, just to keep things simple, you could have drawn the other way around. Remember, for open areas, open surfaces, you can draw either ways as the vector direction, but let me choose this one. My question to you is, and you will see why I'm asking you this question in a second, my question to you is, if this angle is theta, what is this angle going to be? Now, if you have some experience with geometry, you might know that this answer is, you might guess it's either gonna be theta or 90 minus theta, right? So can you just pause the video and think about, just do a little bit of geometry and see what this angle is going to be? All right, hopefully you've tried. So the way I like to do this is, let me take this triangle. If this angle is 90, then I know this angle is nine, sorry, if this angle is theta, I know this angle is 90 minus theta because this whole thing is 90, right? And if this angle is 90 minus theta, and this is a triangle, this is 90, this angle is theta, does that make sense? So this angle should be theta. And corresponding angles, sorry, not corresponding, alternate interior angle. I forget the names. Anyways, if this angle is theta, this angle should also be theta, and therefore we have it. We get this angle to be theta. Now why I asked you to calculate that angle was because now I can say, what is this angle theta? Theta is just the angle between the electric field vector and the area vector. That's it, that's so much easier to communicate. And we can even reduce the words even more because now if you remember your vectors, you might know that when you multiply the magnitude of two vectors and cos of the angle between them, there's a short notation for that. That is a dot product. So we can just say flux is the dot product. You just say E dot A. Like a boss, you can be now a person of few words. This is so much easier to communicate. Flux is E dot A, that's it, done. And even when you're dealing with, say, complex situations where your electric fields and your areas are all oriented in different, different three dimensions and all of that, it's so much easier to take a dot product. When you learn about dot products in maths, you will see it's that easier to do that. It's so much harder to do it this way. And so there you have it. So how do you calculate flux? You do it as just E dot A. Let's just take one last example. So let's say we are given that there is an area of two units, take standard units, which is kept inside an electric field of 100 units at an angle 60 degrees with it. Can you calculate what the flux through that area is going to be? Can you pause and try? All right, my immediate reaction is E A cos theta. So E is just 100, A is two times cos theta where theta is 60 degrees. But this is not correct because remember theta, what is theta? Now we have a much easier definition of theta. It's the angle between the electric field vector and the area vector. So now to calculate the area vector, you have to first draw a perpendicular to it. This perpendicular would represent the area vector. And so now theta should be the angle between this arrow mark and this arrow mark. And that's not 60. In fact, this whole thing is 90 because this is perpendicular. And therefore that angle is actually 30 degrees, 90 minus 30. So this is wrong. It should be cos 90 minus 60, which is 30 degrees. So common mistake that I used to make earlier. So long story short, flux is E dot A. When you're calculating, be careful. Theta is the angle between the electric field vector and the area vector. And how do you calculate direction of the area vector? You just draw perpendicular to your areas. For open surfaces, you can always draw two perpendicular. So it's arbitrary, but for closed surfaces, you can always, our standard is the outward. So for closed surfaces, we choose outward direction as the area vector direction.