Hey, guys. Sometimes you're going to run into problems in which you see vectors represented using a special notation with a bunch of i's and j's, and these are called unit vectors. So in this video, I'm going to show you what these unit vectors are all about and more importantly, how they help us describe and do vector math with vectors a lot more simply. Let's check it out. So, guys, let's think of the easiest way to describe a vector. We have graphical, which is where we have already the grids and the squares. We can take this vector and break it up into its legs and just count up the boxes. Boxes. This is 3 and 4. And then we could use our equations like the Pythagorean theorem to figure out that this is 5 and the angle is 53 degrees. Now, another way we could describe the vector is by already giving the magnitude and the direction like 5 at 53 degrees. Then we'd have to just draw this vector out, so positive x and positive y and then we'd have our 5 meters here at 53 degrees. Now, if we wanted the legs, we'd have to just use our calculating or component equations or decomposition equations to figure out what the legs of the triangle are. A cosine theta, a sine theta, and we just get 34. Now the last way that you'll see vectors represented is using this weird notation with a bunch of i's and j's and k's. These are called unit vectors. And what's going on here, guys, is that these unit vectors are just special kinds of vectors that point in a direction, and they also just have a magnitude or a length of 1. So here's what's going on. If you have a vector, let's say i3+j4, all that's really going on is that I points in the plus x direction, j points in the plus y, and k points in the plus z. So anytime you see i's, j's, and k's, just think of x, y, and z's. Physicists came up with this system a long time ago. They thought it was, you know, they thought you didn't have enough confusing letters so they decided to throw a bunch of i's, j's, and k's in there. So for example, if we've got 3 I and 4 j, all it's really saying is go 3 in the I direction. So we've got 1, 2, 3. So that's 3 of them right here and then go 4 in the j or the y direction. So from here, we're gonna go 1, 2, 3, 4, and this is j j j j and you're gonna have 4 of them. That's all that's going on. So, if you wanted to construct a vector, this is just 34 and so our resultant is just gonna be from tip to tail like this and this is gonna be 5 because this is basically already giving us the legs of the triangle, 3 in the x-x and 4 in the y. So our magnitude's 5 and our angle is 53 degrees. So guys, all of these things here are just different ways to describe the exact same vector. So that's what's going on here. That's all unit vectors are. So you can think about these i's and j's as basically just already being the legs of the triangle. That's what these things are telling you.

Alright, guys. So what unit vectors are really, really helpful for is making vector addition super straightforward. Let's check it out using this example here. We've got these 2 vectors a and b. We’re going to draw them and calculate these, the resultant in unit vector form. So I've got i4+j2. So, basically, I'm going to go 1, 2, 3, 4 in the x direction. I is going to be x and then 2 in the in the j or the y direction. So 1 and then 2. So this right here gives us my a and this here is my ax and my ay. Right? It's just the components of this vector here, but I have it in terms of unit vectors. So this is just 4 I and this is just 2 j. Now, let's do the same thing for b. B is negative I. So if positive I points in the positive x direction, negative I is going to point to the left or the negative x direction. So we've got one to the left and we got 2 up. So we're going to go 1 and then 2. So this right here is going to be my BX and this is going to be my BY, and then the vector is just going to point from start to finish like this. Now, if we wanted to do vector addition, we'd have to follow all the steps. We'd have to make the table and all that stuff and you have to decompose these things and then add them together. But unit vector addition makes this stuff super straightforward. So for example, if we had this a vector, we can just describe it in terms of its I and j components or its xx and y components and I could just say that this vector is, oops, is i4+j2. And then vector b is going to be negative i+j2. So, if we wanted to find the resultant vector, which is the addition of a and b, I already have my x and my y components. And remember that when we have the x and y components, we just add them downwards. We just add them vertically in the table. This is basically already doing that for us. So my a plus b, when we could think about this, is I'm just doing ax+bx and this is going to be in the new x direction and then I'm going to do my ay plus my BY and that's going to be in the j direction or the new y direction. So for example, this new resultant vector is, if I want the resultant vector in unit vector form, I'm just going to do 4 I plus negative J or negative I or negative one I and that's going to be in the new I direction plus now I've got 2 plus 2. So 2 plus 2 in the j direction. So I've got i3+j4. And so, what that would look like, my resultant vector, is I would go 3 in the I direction and then 4 up. So my new vector would look like this. This would be my resultant vector. And this makes perfect sense because if I were to add these 2 vectors, A and B together tip to tail, I'd have to move the B vector over this way. It's one to the left and 2 up and it would get the exact same direction. So basically I would get the same exact vector. So that's just another way to describe, a vector using unit vector components.

Alright, guys. That's it for this one. Let me know if you have any questions.