Math - Introduction to Transformations / Points

 Hi, today I'm going to tell(well, in reality, I'm typing) an introduction to transformations in the Cartesian Plane, but first understanding the point.

First, we need to understand what is a Cartesian Plane.

Simplifying a Cartesian Plane usually is a checkered 2d plane that has a 'y' line(y-axis) and an 'x' line(x-axis), they intersect perpendicular to each other.

Here's an example: 

And we are going to work with this for now.

To work with the Cartesian Plane we need to know the "basic unit" used inside of it.

This basic unit is called a point, yeah, a simple point, but, we can make literally infinite forms with it, but because it only has one coordinate, we call it a 0D form.

First let's understand what a point has, since he is inside the Cartesian plane he has an x and y coordinate. 

We can see in this image that the green point is in the position y 3 and x 2, we represent this as: (2, 3), the x always comes first.

But we can't always call this point the green point, a point needs a label to it, this label is usually a single uppercase letter, like A, B, C, etc.

So let's label this green point as A, to represent it we can write: A(2, 3).

Basically, we can represent a point as Label(x, y).

Making it look like this: 

Also another thing that is interesting to know and can help you in the future is the opposite of a point.

Assuming N is the number you want to find the opposite of:

If N < 0: opposite = N + (N*2)

Else If N > 0: opposite = N - (N*2)

Else If N = 0: opposite = 0

But where do we use it?

This is important for example when performing reflections across the x-axis or/and y-axis because to perform then you need to find the opposite number.

Another ability that's good to have is to estimate the measurement of angles, so here are some common angles that you can find:

Now, let's see how can we calculate the distance between two points.

For example the distance between point A and point B:

This can look hard, but remember about the Theorem of Pythagoras?

If you don't remember, the theorem says that "The square on the hypotenuse is equal to the sum of the squares on the other two sides", so we can create a right triangle to find the distance, you just need to add a new point C that has the same y as point B and the same x as point A, and then we create a right triangle, with this we just need to use the theorem to find the distance, or, the hypotenuse.

This is the basic about points.