Welcome back, everyone. So in recent videos, we've been talking about vectors and various things associated with vectors, like their magnitude and direction. Now, what we're going to be learning about in this video is a new way that we can represent vectors using something called I and j notation. Now if this sounds scary or like it's some kind of crazy code word, don't sweat it because it turns out I and j notation is actually very straightforward. This is definitely a skill you're going to need to have in this course. So without further ado, let's get right into things.

Now let's say we have some kind of vector, vector v. And we're asked to represent this vector using I and j notation. Well, to do this, you first need to understand what a unit vector is. A unit vector is a vector that has a magnitude of 1, and points in any direction. So you can kind of think of it like a unit vector is the opposite of a scalar, where a scalar really only tells us a magnitude, a unit vector really only tells us a direction. It turns out that these unit vectors that point specifically in the x and y direction, we give them special names. The unit vector that points in the x direction, we call this vector I hat, and the unit vector that points in the y direction, we call j hat. Depending on what textbook you use, it's also possible that you'll see these vectors written as x hat and y hat. Now if you ever see this, just know that these mean the exact same thing. But for this video, we're gonna use I and j, but it's also possible you see it as x hat and y hat.

Going back to this example that we had where we had this vector down here. If we want to represent this using I and j notation, well, we can just use this idea of scaling vectors and using the tip-to-tail method that we've already learned about. So if I'm looking at my I vector, which I know points in the x direction, it has a magnitude of 1, I can see that I'm going to need 1, 2, 3, 4 I vectors to get to our vector v. And I can see that I'm also going to need 1, 2, 3 j vectors. So I need 4 I hat vectors and 3 j hat vectors to get this resultant vector, vector v. And that right there is how you can use this I j notation. This right here is the solution. So, really, I and j notation is just asking us what combination of I and j vectors do we need to get whatever vector we're looking for.

It turns out, you might also be given this vector in component form, and component form is something that we've been using throughout these videos. If you're given these vectors in component form, there's a very straightforward way of finding the I j notation. All you need to do is take your x component and multiply it by I hat, and then take your y component and multiply it by j hat. So if we have this vector here, which is 4 comma 3, you could just write this as 4 I plus 3 j. And as you can see, that's exactly what we did here. So this is the idea of I j notation and how to convert from component form to I j notation. And it turns out, there actually are situations where this I j notation is going to be more useful than component form. But to understand this, as well as this general concept of I and j notation, let's get some more practice using some examples.

So in these examples that we have down here, we're given these 2 vectors, vector u, which is 2 comma 4, and vector v, which is 1 comma 0. Now we're asked to rewrite these vectors using I and j notation, and we're going to start with example a. Example a asks us to find vector u, and I can see that vector u is 2 comma 4. So using I j notation, we're going to have 2 i+4j. And that right there is the solution. That's all there is to it. That's example a, right there. Now let's go ahead and try example b. Example b asks us to find vector v. I can see vector v is 1 comma 0, so using I j notation, we're gonna have 1 I plus 0 j. Now I could say that this is my final answer, but it turns out I can simplify things further. 1 times I is just I, and 0 times j is 0. I plus 0 is just I, so vector v is I, and that's the solution. And this is a situation where the I and j notation can actually be more useful than component form. Because notice how we were able to take a vector in component form and we were able to compress it down and write it in a much more simple way using this new notation.

Now let's go ahead and try example c. Example c asks us to find vector u plus vector v. Now vector u, I can see we already figured out. The I j notation is 2 i+4j, and vector v is just, well, I. Now how exactly could we add these vectors together? Well, it turns out using I j notation for any operation is the same idea as using component form. All you want to do is add, subtract, or multiply the individual components. So going back up to this example, well, what I can see here is that we have 2i and we have I. So adding these together, we'll get 3i. And I can see that we have 4j, and then we don't have any j's on this side, so we're just gonna be left with 4j. So 3i+4j would be the solution to example c. So as you can see it's very straightforward. It's just like what we learned for component form.

But now let's go ahead and try example d. Example d asked us to find the u minus 2v. Now I'm first going to figure out what vector u is, and I can see that that's what we already calculated. It's 2 i+4j. Now to find vector 2v, well, 2v is just gonna be 2 times vector v. And you can see vector v is just I, so it's gonna be 2 times I, which could also just be written as 2 I. So now that we have vector u and vector 2 v, to find u minus 2v, well, we can just subtract these vectors we figured out. So we're going to have vector u, which is 2i+4j, and this is all going to be minus vector 2v, which is 2i. Now what I can do is I can subtract the like components. So we're going to have 2 I minus 2 I, which is 0. And then we're going to have 4 j minus nothing, which is just 4 j. So we have 0+4j, which is just 4j. So this right here is the solution to example d. And as you can see, we were able to get a more compact simple.vector by using I and j notation. So this is really what I and j notation is all about, and how you can do some basic operations. So hope you found this video helpful. Thanks for watching.