Furry teaches you the Vector - Linear Algebra

Added: 2026-05-05

[00:00:00]Hello everyone, Mathcat here. Today we will be discussing vectors.

[00:00:06]So what is a vector?

[00:00:09]Well, I personally believe that it's very difficult to explain because looking at different contexts, vectors mean different things. If I could generalize the idea of vectors, I would say that vectors describe something or lack of a better term.

[00:00:33]So let's say you're a computer science student or a data analyst.

[00:00:38]Um this list of data here describes a video. So like let's say you have a video and it has 20 likes, 86 views and three shares.

[00:00:54]This vector is a list of data that um has a relationship with like a video.

[00:01:04]And notice that if you swap any quantities, like if you swap this 86 with this 20, you get 86 likes, 20 views, and three shares.

[00:01:18]Um, that's not possible. But anyways, you get the idea that if you swap quantities, you change the information.

[00:01:28]So now um taking a look in how vectors are described in physics. In physics, vectors are arrows that represent magnitude and direction. So let's say you're throwing a ball like let's say you're facing forward. You're throwing a ball this way. This has direction and you throw it at a certain force. So it has length or magnitude.

[00:01:58]Uh so you can see that if when you throw the ball behind you um you're throwing it very weakly or with less force than as if you're throwing it in front of you or if you're throwing it to the side.

[00:02:14]Um you'll see that the magnitude is still small.

[00:02:18]So the larger the magnitude the more force you're putting into the row. Or another way to describe it is velocity.

[00:02:27]Since velocity is a vector, you can measure speed as a magnitude um but it doesn't have direction. And you can measure like direction as a direction but it doesn't have any magnitude. But if you have a speed that's going a certain direction, then that is velocity.

[00:02:50]That is the definition of velocity.

[00:02:53]And in linear algebra, we describe vectors as kind of like instructions.

[00:03:00]So let's say you have a vector one one.

[00:03:08]This is describing like the um movements.

[00:03:16]So notice how the list of data corresponds to like certain variables.

[00:03:24]What this is telling us is it's giving us instructions for how to move. So it's saying move one in the x direction and move one in the y direction.

[00:03:39]And if we define X and Y then we can see that this vector corresponds to this numerical representation.

[00:03:53]So as you can see vectors describe many different things. So that is why I say that they describe something. It describes an idea.

[00:04:07]So let's move on to um more linear algebra based representation of vectors.

[00:04:20]So in linear algebra vectors will almost always um start from the origin. So let's say you have a coordinate plane. All right.

[00:04:35]And then you u make arbitrary trick marks. Tick marks describing distance.

[00:04:51]This is the x direction and this is the y direction.

[00:04:56]So let's say you have a vector 1 2 and you have another vector.

[00:05:09]You have another vector.

[00:05:16]Vectors will always start at the origin.

[00:05:22]So um let's describe this blue vector.

[00:05:27]It is telling us to move um let me write the instructions or the variables. So it's telling us to move one time in the x direction and two times in the y direction.

[00:05:45]So one time in the x direction and two times in the y direction.

[00:05:53]And this kind of move movement this can be simplified down into an arrow that is curved for some reason. um it can represent an arrow that's located at that location. So like if you want to think about it this way, think of these as ordered pairs and you can represent this vector as an arrow pointing to this coordinate.

[00:06:34]And you can do the same thing for here.

[00:06:37]um to describe all the vectors. This is telling us to move one time in the negative x and one time in the positive y.

[00:06:55]So there would be a point here. And to visualize this as a vector, we connect the origin to this point.

[00:07:06]And then we do the same thing for this green vector.

[00:07:10]So move twice in the x direction, move twice in the y. You'll have a point here.

[00:07:21]And then you you will represent this as a vector or an arrow pointing to that coordinate.

[00:07:34]Notice how they all start at something called the origin which is at 0 0 or the middle of this entire coordinate plane.

[00:07:45]So just by looking at this we can describe some of the properties of the arrow like the magnitude like how long this arrow is. We can see this arrow is kind of short and this arrow is kind of long. Can describe the length or otherwise known as the magnitude the magnitude of a vector.

[00:08:17]If you remember the Pythagorean theorem, the magnitude of the vector, this um long part can be described with the instructions that build that specific vector. So um we can imagine a triangle leading up to this. So move once in the X, move twice in the Y and you can see that it forms a right triangle. And then to find that magnitude, all you would need to do is to perform the Pythagorean theorem.

[00:08:57]So this horizontal component walking once to the X represents the actual like a^2 and then this represents B^2.

[00:09:10]And then this blue vector is c^ 2.

[00:09:14]So using the instructions of the blue vector 1 2 + 2^ 2 is = c^ 2 and c is equal to the square t of 1 2 + 2 2.

[00:09:37]This simplifies down to square<unk> of 5.

[00:09:43]So the magnitude of the blue vector would be the square<unk> of 5.

[00:09:50]And the same thing can be done for the red vector to describe its magnitude/length um square<unk> of -1 + 1.

[00:10:06]And these are both squared. This is equal to square root of two.

[00:10:14]And then same thing can be done for the green vector.

[00:10:21]So 4 + 4<unk> 8. So these three represent the magnitudes of each of the vectors or its length.

[00:10:34]And then one thing about magnitudes is when you find the magnitude of a vector, magnitude can be represented as double bars around a vector.

[00:10:57]This is equal to um as we saw earlier the magnitude of square root of 8. If we divide this vector by the magnitude, we get something called the unit vector.

[00:11:15]And let's say the original magnitude looked like this.

[00:11:27]If this um draw it somewhere else, this green vector represented the green vector and had a magnitude of this. Then the unit vector would be in the same direction but it would have a magnitude of one. So if you divide the magnitude or divide the vector by its magnitude, you get something called the unit vector which is very useful in things like proje um not projections but actually yeah projections when you um don't really care about the magnitude of a vector.

[00:12:19]And then to find the direction you can utilize the same properties.

[00:12:27]So you know the instructions that lead up to the vectors.

[00:12:35]You can just imagine it as a triangle side length two in the x and y.

[00:12:45]And then to find the angle relative to this coordinate basis, we want to find this angle here.

[00:12:54]Then we define this as theta using imagine you didn't know the magnitude using what you know we can use tangent theta is equal to do you remember this um opposite over adjacent because of soa.

[00:13:21]So to opposite over adjacent then we can take the inverse of both sides to find theta.

[00:13:38]So we'll just get theta is equal to the tang or the inverse of tangent of opposite over adjacent which would be arc tangent if you'd like to describe it as that but we'll just keep it as inverse of tangent.

[00:14:01]So this angle is equal to the inverse tangent of opposite over adjacent which would just be 2 / 2. And that simplifies to inverse tangent of 1. And then you can plug this into your calculator and then the angle would be 45°.

[00:14:35]So this angle relative to the x axis is 45°.

[00:14:44]The same thing can be done for this vector and this vector and that will define the direction and note that this is in 2D.

[00:15:00]Um we can find the angle in 3D, 4D, 5D by using their respective methods but this method works for 2D dimensions.

[00:15:10]Also, if we were given the a instructions in kind of like the reverse order, um like we were given direction first, which is 45° relative to the x-axis and then the magnitude of um the square<unk> of 8, it will result in the same vector.

[00:15:36]And that is also kind of the idea behind how vectors are represented in physics.

[00:15:43]Direction and magnitude. There are some properties of vectors that I would like to talk about and that would be vector addition and vector subtraction as well as scalar multiplication and that will cover all three.

[00:16:01]So let's say again you have a coordinate plane X Y with tick marks with an arbitrary length.

[00:16:25]Okay, that kind of looks ugly, but whatever. Um, so let's say you have a vector 2 one and then you add let's make this one one and then you add um one one if If you read out the instructions, this is telling you, let's follow the path.

[00:17:07]So, um, let's start with this blue vector.

[00:17:12]You move once, let me make this brighter. You move once in the x direction, once in the y direction, and then you add the green vector.

[00:17:28]So then you take a step back in the x direction, but then you add one more time in y.

[00:17:37]This would result in the red vector.

[00:17:43]So um to add these just add them to their correspond add the elements to their corresponding um row. So like for this top row, imagine there's like kind of like a separation. This is kind of like adding the like terms.

[00:18:03]Can visualize x and y's here.

[00:18:06]So um adding like terms we get one adding negative 1 which is zero x.

[00:18:19]And then doing the same thing here. One Y add one Y that's two Y. So you move twice.

[00:18:29]So then you have the red vector. And following these instructions X and Y. So you move zero in the X direction, go up twice in the Y direction. And you can see that the combination of these steps ended up at this point.

[00:18:51]And the red vector describes that very same point.

[00:18:58]So you can draw an arrow here.

[00:19:01]And this is vector addition.

[00:19:05]The same thing happens for subtraction.

[00:19:09]So if you decide to subtract the vectors you get something like um so you do the same instructions you move once in there and then you subtract one in the x direction. So then that is just adding one. You go once in the x direction and you subtract one in the y direction.

[00:19:47]So then you end up this point that can be represented with a yellow vector.

[00:19:58]There's this line right here.

[00:20:04]So, um, if you've ever heard of the term like tiptotail addition, if you add a vector, um, and you have a vector u and v, let's say vector v is here and then you add vector u. So you walk here, then you add another vector. So you walk another direction. These are just random vectors. And the sum of these two vectors will equal let's call it vector E.

[00:20:51]You'll go from the start of the first vector and end at the second vector.

[00:20:58]So um again visualizing in the steps let's say you only took um you did all the horizontal components first and you did the y components.

[00:21:15]So uh let's do it with purple.

[00:21:22]Let's do the first step first.

[00:21:25]You move here as this whole vector to represent this vector and then you add this vector starting at this point.

[00:21:40]Then you would get one. So you go down one or go back one and you go up one.

[00:21:49]So then you would end up with this red vector because you add up the tips and tails of the vectors. You start here, end up at vector one, then from there add vector 2, you get the resultant vector.

[00:22:11]So when I said that um vectors will almost always start at the origin that is um that doesn't apply for vector addition and vector subtraction because let's say if you follow the instructions like you walk once in these directions and you want to add a vector it wouldn't make sense to start back from the beginning.

[00:22:40]Like if you're walking here and then you add another set of instructions like you walk up here, it wouldn't make sense to walk there and then go back and then walk that direction.

[00:22:55]So that is what people mean when they say tiptotail addition.

[00:23:02]And then same thing applies when you're scaling vectors.

[00:23:07]So if you let's have a coordinate plane here and let's define a vector u.

[00:23:18]Um if you scale it by two then it is as simple as scaling this vector by a factor of two.

[00:23:33]you kind of increase that vector as if there were two of the little vectors.

[00:23:44]So it increases in size or it scales. So let's say you have negative u you just flip it. Flip the direction.

[00:24:01]And this represents negative U. If you wanted to scale it, let's do yellow.

[00:24:08]First, you'd want to flip it and then scale it by a factor of two.

[00:24:24]All right. And that should help you form the basis of what the vectors actually mean when you're taking a look at matrices. All right. So even though the computational aspect of vectors and the idea I believe that um it's very easy to learn but trying to describe the abstract idea of vectors was definitely very difficult and it's difficult coming from someone who has already learned and went through and understood vectors but I'm not sure if that was able to capture or help anyone who wasn't able to understand vectors previously to now actually have an idea. Um, I'd say that this is one of my most difficult videos.

[00:25:20]Um, but as you can see here, we have some fan art. And just a reminder, if you would like to submit fan art, um, I have a Google form in the description to submit fan art. And please submit your fan art.

[00:25:37]I believe that no art is bad and everyone is at their own pace and I love all fan art except for AI art. Please do not send me that. Um, but if you would like to also support me and help fund my college education, I have my Kofi in the description.

[00:26:00]And oh, also I'd like to say that I'll display three fan arts per video. So if your fan art isn't displayed yet, please don't like DM me and ask me, "Hey, why is my fan art not being displayed?" Um, because I think this is the best format.

[00:26:20]Unless I start getting too much, then I might have to reconsider. But I think three is a good number for now.

[00:26:27]Oh, thank you for watching. And as a reminder, I have some extra art and lecture notes available on my Instagram and Tik Tok.