Vector

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Transcript

Before you understand how you can process images by machine, it's important to understand the concept of a vector. Okay? So let's do that now. So a vector is nothing but that has some big magnitude and directions. So let me illustrate that with a diagram. So So let's say this is a vector.

So vector has a tell and it has a head. So this is known as the magnitude, which is the length of the vector. And this is known as the interaction selector is described by two things. magnitude and a direction, right? So let's understand this example with an example of a coordinate system. So let's say let's start with the two dimensions first.

This is x, and this is y. So let's say you have two points here are the points here, right and the line connecting them. Let's say it's also passing through an origin. And the line has some scores, slope and some height right and y intercept. So this line can be represented by a vector. Let's say the equation of this line is y is equal to mx plus edge.

So M is a slope and it is a height. And you can represent this has a vector which is to have two numbers m Ah, so this is how vectors vectors usually just refer to the list of numbers in any coding language. So in this particular case, it's a two dimensional vectors. And you can represent as a list of two numbers, M and H. Right. Let's take another example a little bit. in three dimensions, what happens?

So let's say this is, this is x axis is y. And this is z axis rate. And let's say you have a point here. So this point can be represented as. This is the length along x axis, let's call it as I. This is length along y axis.

Let's call it as a y And this is lentil z x is called a ski. So i j and k this will have usually say this is five minutes long, this is six units long and this is three units long. So, this vector point vector can be described by numbers, which is either a case of five, six and three in notation is described as five i plus 60 plus three. Okay. So, this is how you represent vectors in machine learning. Then the next thing you understand need to understand is how you can represent a similarity between the two vectors, right.

So, you have a one vector In particular, in other words, another particular action, how can you compute the similarity between the two vectors? So there are many ways to calculate the distance between the two vectors. What is the Euclidean distance, then you have cosine similarity, Manhattan distance in Jakarta and many more, we cannot go through all of those similar distances in this lecture. But let's explore a couple of them. Right? Let's try example of Euclidean distance.

Euclidean distance is nothing but an extension of a Pythagoras theorem where it so this is a coordinate system. And let's say this is one vector and another vector, right? So it represents absolute distance between them. So let's say this is having a coordinate x i and y i. This is having a coordinate x i YI again. So the Euclidean distance between these two vectors is represented as summation of x minus x squared plus y i was minus y i two square.

That's the distance you calculate, using Euclidean distances. This is Euclidean distance. Right? Then the other thing that you can learn about cosine similarity. So cosine similarity indicates the angle between the two. So let's say this is one vector, v one, and this is another vector, v two.

So this is the angle between them, right? It's called t de. So the similarity between the two vectors is the cosine of angle between the two. So, so here if the angles are if the cosine similarity is a measure of direction and not the magnitude. So let's say I draw another vector, let's say called v3. And here is another angle, let's say theta two theta one.

So three three is more closer to v one and the cosine of that angle would be how much so it will be less it will be lesser as compared to cosine of data one because cosine of zero is one. Cosine of 90 is zero. Great. So if the vectors are very close to each other, that means the similarity cosine of them will be one and as you move away, the coastline will reduce to zero Right vectors are said to be orthogonal when they are at 90 degrees and the v1 and v2 triangle is entity and they are very dissimilar to each other and they are also called orthogonal. So, the distance of cosine distance and the cosine distance the lower is angle. The vectors are more similar in case of Euclidean distance in just a Pythagoras despite the average distance between the two vectors, and lesser is the distance, the more closer the vectors are.

So, that's how vector similarity is calculated.

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