Lesson Video: Horizontal and Vertical Components of Vectors Mathematics

Video Transcript

We already know that a vector is a set of numbers that can be represented in a suitable space by a line segment with a specific length and direction. We’ve also seen that a line segment has a magnitude and direction, which basically means that we can describe it by saying how long it is and in which direction it’s pointing. In this video, we’re gonna talk about horizontal and vertical components of two-dimensional vectors and introduce the 𝑖 and 𝑗 unit vector notation.

Any two-dimensional vector has two components. The first represents the amount of movement in the 𝑥-direction and the second, the amount of movement in the 𝑦-direction. Of course, when I say movement, I’m just talking about differences in 𝑥- and 𝑦-coordinates in a graphical representation of the vector by a line segment on a graph. The vector itself might be representing something completely different, force or acceleration, for example. So in this case, the amount of 𝑥-movement is this distance here. So we’re going from an 𝑥-coordinate of three to an 𝑥-coordinate of six, so that’s a difference of plus three. So our 𝑥-component of the vector is positive three. The 𝑦-component is this bit here. The 𝑦-coordinate of 𝐴 was two, the 𝑦-coordinate of 𝐵 was nine, so that’s a difference of plus seven. And we can write that as 𝐴𝐵, vector 𝐴𝐵, in this format here: three, seven. So the three is the 𝑥-component and the seven is the 𝑦-component.

And if we had a three-dimensional coordinate system with 𝑥- and 𝑦- and 𝑧-coordinates, we would just be inserting another number at the end here onto our vector. So we can extend this system to any number of dimensions.

We define two special vectors, 𝑖 and 𝑗, to be positive one in the 𝑥-direction or positive one in the 𝑦-direction, respectively. So this is 𝑖, it’s just a movement of one in the 𝑥-direction. And here’s 𝑗, a movement of one in the 𝑦-direction. So remember that the 𝑖 and 𝑗 vectors — are at one, zero and zero, one — can be placed anywhere on the graph. They don’t have to start at the origin. So there we are; I placed them somewhere else. Each vector is just describing a particular journey. In this particular case here, for 𝑖, we’re adding one to the 𝑥-coordinate and we’re leaving the 𝑦-coordinate as it is. So we’re doing this journey from here to here. In the 𝑗 case, we’re not adding anything to the 𝑥-coordinate, but we’re adding one to the 𝑦-coordinate. It describes this movement from here up to here.

Now remember, we can also use the rules of adding and subtracting vectors to stack up these 𝑖s and 𝑗s, to create bigger vectors. So, for example, this vector here, 𝑖, represents a movement of one in the 𝑥-direction and none in the 𝑦-direction. If I increase that to this vector here, so this whole length here, that would be two 𝑖s following on from each other. Or, it will be a translation of two in the 𝑥-direction and zero in the 𝑦-direction. Now if I added on to that this vector here — which starts at the end here and then moves up not one, not two, but three — that would be three 𝑗 vectors added on together, making a vector of three 𝑗 or zero, three. So if I add two 𝑖 plus three 𝑗, that represents this green journey from the start point to the end point up here. So two 𝑖, the 𝑥-component would be two. Three 𝑗, the 𝑦-component would be three. And the direct journey two 𝑖 plus three 𝑗 is the green line. So this means the 𝑥-component was two and the 𝑦-component was three. So we’ve now got two different ways of representing this green vector here. We can use it in the standard vector notation that we’re familiar with. But we’ve got this new notation here, in terms of 𝑖 and 𝑗 vectors. The number of steps in the 𝑥-direction is the 𝑖s and the number of steps in the 𝑦-direction is the 𝑗s.

So let’s sum that up in the general case. If we start off at point 𝐶 here with coordinates 𝑥 one, 𝑦 one and we end up at point 𝐷 here with coordinates 𝑥 two, 𝑦 two, then vector 𝐶𝐷 is, the 𝑥-component here is the difference in the 𝑥-coordinates and the 𝑦-component here is the difference in the 𝑦-coordinates. So that’s our standard vector way of writing it. Or, we can say that that is 𝑥 two minus 𝑥 one lots of the 𝑖 unit vector plus 𝑦 two minus 𝑦 one lots of the 𝑗 unit vector. That make some kind of sense. All we’re saying is that the 𝑥-component of a vector is the coefficient of 𝑖, in this format, and the 𝑦-component of the vector is the coefficient of 𝑗, in this format. So we’ve just got this new notation where we have the 𝑖 vector, which is a step of one in the 𝑥-direction, the 𝑗 vector, which is a step of one in the 𝑦-direction, and we just say how many of those we’re doing in each case to make up our vector.

Right. So now we know this new format. Let’s just have a look at a couple of quick questions that involve using 𝑖s and 𝑗s in our questions. So we’re going to do this question here.

We’ve got 𝐴𝐵; it’s the vector three 𝑖 plus four 𝑗. 𝑋𝑌 is the vector negative two 𝑖 plus three 𝑗. And we’ve just got to add those two vectors together.

So the first stage is just to take vector 𝐴𝐵 and add vector 𝑋𝑌. And all we have to do is add the 𝑖s together first, and then add the 𝑗s together second. So three 𝑖 add negative two 𝑖 is just one 𝑖, so that’s 𝑖. And four 𝑗 add positive three 𝑗 is seven 𝑗. So there’s our answer, 𝑖 plus seven 𝑗. Simple as that, add the 𝑖-components together, add the 𝑗-components together, look out for the negative signs and- when you’re doing those calculations; but otherwise, that’s a pretty straightforward process.

So let’s just visualise that example. So we had three 𝑖 plus four 𝑗. So we’ll be going positive three 𝑖 and then positive four 𝑗, this is the 𝑎𝑏 vector. So there we are. We’ve just laid that down at the origin. We could’ve laid it anywhere on the-on the graph. Now adding vectors, we just lay them end-to-end. So what we were adding, 𝑥𝑦, was negative two 𝑖 plus three 𝑗. So we’re effectually starting 𝑥 from 𝐵, so 𝑥 lays on top of 𝐵 and we’re going negative two 𝑖. So we’re going two in the negative 𝑥-direction and we’re going positive three 𝑗, up to here.

So adding vectors is just a matter of laying them end-to-end on the graph. So we laid 𝐴𝐵 down, which started here and ended here, and then we just added 𝑋𝑌 to the end of that, laid that onto the end. So that started from where we just finished off and then ended up here. So the resultant vector is this green one here. And to get from the beginning of the green vector to the end of the green vector, we had to go positive one in the 𝑥-direction. So that’s one 𝑖, or just 𝑖. And in the 𝑦-direction, we’re going up seven all the way up here. So that’s plus seven 𝑗.

So when you’re doing these questions, it really is just a matter of adding the 𝑥-components together, adding the 𝑦-components together, and coming up with a simple answer. You don’t need to do all this graphical checking. But I’m just hoping that that’s giving you some extra insight into the process and why it works.

Right. Let’s take a look at one final question then.

We’ve got vector 𝐴𝐵, is three 𝑖 take away four 𝑗. Vector 𝑋𝑌 is negative four 𝑖 add seven 𝑗. And we’ve got to find vector 𝐴𝐵 take away vector 𝑋𝑌.

So just writing that out, 𝐴𝐵 is three 𝑖 take away four 𝑗. 𝑋𝑌 is negative four 𝑖 plus seven 𝑗, and that’s what we’re taking away from vector 𝐴𝐵. So we just need to be really careful, when we’re taking these away, about the signs here because we’ve got a negative outside the bracket, we’ve got negatives and positive inside the bracket.

So let’s just start off with the 𝑖-components then. I’ve got three 𝑖 and I’m taking away negative four 𝑖, so that means I’m adding four 𝑖. So three 𝑖 add four 𝑖 is seven 𝑖. And then for the 𝑗-components, I’m starting off with negative four 𝑗 and I’m taking away positive seven 𝑗, so f- negative four take away another seven is negative eleven 𝑗. So there’s our answer, seven 𝑖 take away eleven 𝑗.

And just to summarise then what we’ve learnt. 𝑖 is the unit vector in the 𝑥-direction one, zero and 𝑗 is the unit vector in the 𝑦-direction zero, one. And given any vector — like 𝐴𝐵 is five, three — this here is the 𝑥-component, this here is the 𝑦-component. We can rewrite this as five 𝑖, because that’s the number of 𝑥s, plus three 𝑗, because that’s the number of 𝑦s. And given two vectors, 𝐴𝐵 and 𝐶𝐷, with their 𝑖- and 𝑗-components like this, in this format, we can add or subtract them just by adding or subtracting their 𝑖-components and their 𝑗-components, separately.

So, for example, adding 𝐴𝐵 and 𝐶𝐷, we can add the two and the negative two 𝑖 and we can add the negative three and the four 𝑗. Well, in this case, two plus negative two 𝑖, that’s zero 𝑖s, so we don’t need to bother writing zero 𝑖. And negative three plus four is just positive one, so we end up with an answer of just 𝑗, one 𝑗. And if we want to subtract the vectors 𝐴𝐵 minus 𝐶𝐷, we’ve got two of the 𝑖s here take away negative two 𝑖s, and we had negative three take away four of the 𝑗s. Then two take away negative two is two add two, so that’s four of the 𝑖s. And negative three take away another four is negative seven 𝑗s. So we’re adding negative seven 𝑗. So we probably not write plus negative 𝑗, we just write four 𝑖 take away seven 𝑗.

So hopefully, you’ll be comfortable using the 𝑖 and 𝑗 unit vectors just to represent the 𝑥- and the 𝑦-component of any vectors that you come across now.