 When working with vectors in algebraic form, especially in physics, it's very common to represent the vectors using unit vectors. What's a unit vector? A unit vector is a vector whose magnitude is exactly one, hence unit. And there's two particular unit vectors that we use in this context. There's the unit vector i and j, which of course, if you write them with the computer, you write this in boldface font. But when you write it by hand, it's hard to write things bold. So you draw a little arrow over them so they look like vectors. Now, i and j should have a little dot on the top of it in iota. Oftentimes, you write the unit vector i and j. You omit the dot because the arrow would kind of go through it anyways. And so you see this i and j without their iota, or I should say their iota is replaced with this vector arrow. The vector i is commonly referred to as the unit horizontal vector, and then j is the unit vertical vector. And they're defined by the following formulas. i is the vector which is 1 in the horizontal direction, 0 in the vertical, and then j is the opposite. j is 0 in the horizontal direction, it's 1 in the vertical direction. If you were working with three-dimensional vectors, you would then introduce a third unit vector k, which k then would be it's 1 in the z direction, it's 0 in the x and y direction. But again, we won't really worry about three-dimensional vectors here whatsoever. Now, be aware that if you're given any vector in algebraic form, v equals a comma b. This can be rewritten as v equals a i plus b j. And I want you to see why that is. If I take a i plus b j, just this linear combination here, well, what is i? i is 1, 0. What is j? It's 0, 1. If we do the scalar multiplication, we end up with a 0 plus, and then with the second one, you're going to get 0b, you add that together, you end up with a comma b, which was v, right? So every vector can be very easily expressed in this form right here. And so this translation seems a little bit lengthy, but it's very common. Again, especially in physics settings, physicists love to write vectors in this form, v equals a i plus b j, all right? So if we had a vector like v equals 3, 4 right here, how do you write in terms of these unit vectors? Well, this would look like 3i plus 4j. That's all there is to it. You just take the horizontal component, that's the coefficient of i, and then you take the vertical component that becomes the coefficient of j. And then here's a diagram to see what's going on here. The horizontal component is just going to be a scalar multiple of i, right? So with this vector, its horizontal component is 3i, and its vertical components can be 4j. That's all there is to this unit vector notation here. So imagine we have a vector v, and in standard position, let's put in standard position, that's to say, it's going to make an angle of 35 degrees with the positive x-axis, all right? So we're already seeing a picture that looks something like the following. Our vector, it looks something like this, it forms this angle of 35 degrees with the positive x-axis. Let's say its magnitude is 12. Can we write this in terms of the unit vectors i and j? Well, so we know that any vector can be written using unit vectors, so you're going to get ai plus bj. So we need to find these numbers, but a and b are just the horizontal and vertical components of the vector, right? And so a is just the horizontal component, vx that we've introduced previously, which would just be the magnitude times the cosine of this angle theta. And so we end up with 12 times cosine of 35 degrees, 35 degrees. With 35 degrees is not any special angle. I'm going to approximate this and round this to the nearest decimal place. I'd get 9.8b on the other hand, which is the vertical component. This is going to look like the magnitude of v times sine of theta, which in this specific situation we get 12 times sine of 35 degrees. And like I said, 35 is not some special angle we have memorized, so we'll just use a calculator to help us out here. This will give us approximately 6.9, like so. And so then we see that v is equal to 9.8i plus 6.9, 6.9j, like so. And so just recall that this ij notation using the unit vectors is just expressing the horizontal and vertical components of the vector. So hopefully it's clear that expressing v in component form is no different than expressing it using these unit vectors i and j. And so let's look at some calculations, some algebra using these unit vectors. Let's do some linear combinations. Let's suppose that u is given by the vector ui minus 5j. And let's suppose that v is given by the vector negative 3i plus 7j. Let's add the two vectors together. Well, one thing I want to point out to you is when you add these vectors together, like so, well, u is the linear combination 4i minus 5j. And v is the linear combination 3i plus 7j. And when it comes to vector addition and skilling multiplication, these algebraic properties, these operations follow the usual algebraic rules you're used to. Addition is commutative, it's associative, that is we can redo the order, we can do parentheses, not a big deal. We have an additive identity, the zero vector, we have additive inverses, skilling multiplication distributes, all of those usual rules work for us. Therefore, things like combined like terms are applicable when it comes to adding together vectors, combined like terms. In which case, then we could take, for example, 4i minus 3i, put those together, and then you're going to take negative 5j plus 7j. For example, like if an intermediate algebra student were to start watching this video at this moment and they know nothing about vectors, what would they do in such a situation? They'd be like, I combined like terms 4i minus 3i is equal to 1i, all right, and then negative 5j plus 7j is a 2j. And you don't have a coefficient of 1, you don't like to do that. So I'm just going to simplify it to be i plus 2j. So that's how they would do it. And they would actually be right, that's how you add together vectors, you can add like terms. And so one of the advantages of writing vectors using this unit, these unit vectors, as opposed to the brackets we did before, is that it feels more algebraic. That is, it doesn't feel so alien compared to algebra we've done before. If you just treat i and j as variables, we can combine like terms, but they're not really variables because they have meaning. It's not that x is a placeholder for some future number. i and j are directional vectors, but we can still combine them together in the usual way. Now, notice if we approach this problem from a different perspective, if we took u plus v, like so, you end up with 4 negative 5 plus negative 3 and 7. This is how we were doing it before, you add together the x parts, you get 4 negative 3, you add together the y parts negative 5 and 7, you end up with, well, 4 take away, 3 is 1, and then negative 5 plus 7 is 2, like we saw before, 1 and 2. And that's the exact same thing. This would be the vector i plus 2j. So the way we add them together is no different, but using the unit vectors, again, it feels more natural, less alien. It's how we've been adding things together, just combine like terms. And so that's one reason why some people like this i, j notation. It feels more like algebra, but it's doing the same thing. Well, can we compute 3u minus 4v? 3u minus 4v, like so. Well, remembering that u is given as 4i minus 5j, and remembering that v was given as negative 3i plus 7j, what do we do? Well, again, if that intermediate algebra student were to walk in right now, they would be like, okay, let's distribute the 3, let's distribute the negative 4. I'm supposed to simplify this thing. I get 12i minus 15j. Then I'm going to get a plus 12i and then a negative 28j, like so. Then they combine like terms. Okay, I can add the i's together. I can add the j's together. I don't know why they didn't say x and y, because again, the intermediate algebra student's talking right now, he doesn't know vectors. You get 12i plus 12i, which is 24i. And then you're going to get negative 15j minus 28j, which is negative 43j, like so. And that gives you the linear combination 3u minus 4v. In which case scalar multiplication just looks like a distributive property when you have these unit vectors. So working with vector algebra is really not that different than working with the algebra you're used to, especially with these unit vectors. It just kind of feels like I have some variables that have a very specific meaning. And then with that in mind, you can add, subtract, and multiply these things by the usual rules.