Easy Maths: Vectors

Hello there!

Today's blog is about maths, more specifically - the rather interesting side of it and what I have learned. Keep in mind that I am not an expert in the subject, so any kind of criticism is more than welcome. With that being said, let's get into the maths.

Preface

This post will cover the topic of vectors, a topic I really enjoy since I use it a lot in game development. I will show you what they are, how they can be used to draw shapes, finding out the length of them and why multiplying them together gives pretty cool results.

How to think of Vectors

I tend to think of vectors as a point in space away from the centre, think of a line that goes from the centre to that specific point. In terms of adding vectors, you could think of it as taking one string and adding it to the position at the end of the string. Or you could just quite literally add the two x's and y's together. Subtracting the two is quite self-explanatory. That is it. They are just points in space.

The power of Pythagoras'

Anyone who has ever heard of that maths question where you need to find the hypotenuse of a triangle, has probably heard of the term Pythagoras' theorem (or the 'Pythagorean theorem' if you have lived in the United States). The way they do it in school is so impractical and shoe-horned that it would put anyone off the whole thing, couldn't one just use a ruler to measure the hypotenuse? Not all triangles are right angled in the real world, in fact very few of them are.

With that being said, it can be very useful in terms of finding the magnitude of a vector (length of the vector), you plug in the X and the Y (if either are negative, make them positive) into Pythagoras' theorem and you have your magnitude!
Now that we have established that, I'll introduce you to another type of vector which is a unit vector, it has a magnitude of 1 and is useful for wanting to move in a certain direction. The way that this is calculated is by dividing both x and y by the magnitude of the non-unit vector. The result is that the two points would either be no more less than -1 or 1.

If you tested this in a game however, the movement speed would be very slow, therefore you would need to multiply it by a single number i.e. (0.5, 0.9) x 20 = (10, 18).

Geometry

Vectors can also be used to make shapes that have vertices, a good example is a triangle or a polygon. More complex examples can include 3d shapes where each vertex is a point that has variables x,y and z. The way that you can turn a bunch of points into a shape is by drawing a line from one point to another much like those drawing books which have you connecting the dots together. You can scale these if you want by multiplying the vectors by one number (similar to increasing the speed of movement). Multiplying a vector by another vector is not what I mean here, but would nicely lead into the next section.

Dot product

When I first heard of this, it was a seemingly simple idea, multiply 2 vectors together.
The formula for this is:

(X1 x X2) + (Y1 x Y2) = Dot product of V1 and V2. 

It doesn't seem too practical. How can I ever use that for anything useful? That kind of thinking is very mistaken, namely because it can be used for a lot like collision detection. A good analogy I came up with is a rod. A pretty long one, in fact one that stretches to infinity in both negative and positive direction. The direction of the rod is determined by a directional vector, ideally a unit vector, since it is very easy to measure. And there are points around it.

Here's where the interesting stuff comes in, imagine that the points fire lasers perpendicular to the rod and they are so hot it leaves a mark on the rod. To make things simpler, lets rotate this whole rod (along with the points) so they all sit in a line.

Let's get rid of the lasers and just focus on the rod.

All that is there is a rod and the red marks. The very useful thing about this is that you can compare where the 'lasers' are relative to that point using these red marks. You could use this to project a shadow from a shape to the rod using said shapes' vertices.

Conclusion

Vectors are pretty cool in my opinion, in fact they are one of my favorite parts of maths (that was, after I stopped doing compulsory maths and actually learned it 'on my own'), it makes me think about how objects are moved or what causes them to move in the first place.
Even if you never use it, just remember that you can at least view the world in a more interesting way if you know vectors.
That's all from me!

Easy maths: Sin and Cosine

Hello there!

Today's blog is about maths, more specifically - the rather interesting side of it and what I have learned. Keep in mind that I am not an expert in the subject, so any kind of criticism is more than welcome. With that being said, let's get into the maths.

Preface

Sin and Cosine are the concepts you may have first heard when you were being taught how to find a certain side/angle of a triangle. The phrase "SOHCAHTOA" often tends to pop up, and a lot of the questions you were given you just had to use this formula to answer them. As well as explaining what this phrase means, I'll also explain why you use this to find sides ,something your maths teacher probably never explained.

Relations with the Axis

Sine = X and Cosine = Y. I've pretty much told you the answer right there. So, next section....

Actually, I jest.

Look at the triangle below:

It has the hypotenuse, adjacent and opposite side labeled for your convenience, so you don't need to crack open your maths book.

If we want to find, for example the adjacent, we will need to use SOH (Sine, Opposite and Hypotenuse). Why?  Because the adjacent is a horizontal line, and Sine is used for working them out (think: the X axis).
Now, we're not going to do a maths question, so don't panic. I'm just telling you a theoretical example. Anyway, Sine is the thing you would use to work out the horizontal parts of the triangle. As you can imagine you would need Cosine for the Vertical axis (or the opposite in this case), therefore you need to use CAH. I'm not going to talk about TOA because I'm not talking about Tangents in this blog, only Sine and Cosine. But hopefully I've made your maths homework a bit easier. Sin for finding out the horizontal side and Cosine for finding the vertical side.

Okay, at this point in time, you may wonder "Right, I know that Sine is used to work out the horizontal and Cosine for the vertical, but why? What is the relationship between these?"

I'll tell you in the next section.

Angles to Vectors

You may have also encountered angles at some point in your maths career, the things you use a protractor to measure or what you have to figure out in a triangle, on the opposite of the right angle, (using Sine and Cosine). I'll tell you why:

As you can see, in Desmos this is what Sine (Red) and Cosine (Blue) looks like in a graph. It produces quite a wavy pattern, which looks quite nice. This wavy pattern is far more than just a beautiful little pattern, it also shows how Sine and Cosine functions, in the graph, Cosine is a little bit more delayed than Sine in terms of reaching its top and bottom, but by how much exactly, and why?

If you look below at the two diagrams, the top point is 1 and the lowest is -1, the more interesting point is the horizontal point, which is half of PI (3.142...). If you multiply this by 57.672 (to convert radians to degrees), you get a number which is ~90. Sounding familiar?

If you quadruple 90 you get 360, which a full circle is 360 degrees (or 2 PI radians). The relation of this to Sine and Cosine is that the distance that they are apart is equivalent to a quarter of a circle. I'll demonstrate what I mean below:

Make sure the setting is in degrees rather than radians.

As shown, B is 90 and the point on the circle is where 90 degrees would be going, and the number '1' is shown again, much like the previous diagrams. The reason that Sine is used for horizontals and Cosine being used for Verticals is shown in the example, due to the aforementioned separation by half a PI. If we swapped Sine for Cosine, then the point would be on the top rather than the right side and that wouldn't look like 90 degrees. Since these two operate in a wavy pattern where the highest value of Sine and Cosine is 1 and the lowest is -1, they can be a useful way of converting angles into a unit Vector. I hope you are starting to see how Cosine and Sin can be used to be turned into vectors.

This is especially handy if you want to simulate rotations via having degrees rather than vectors. 

Conclusion

This is my first time explaining mathematics to you, and I feel a bit nervous doing so, since I don't really have much of a formal background in the subject beyond GCSE. But as I've stated before I welcome any form of criticism.

Sine and Cosine are quite useful things, probably more than you thought - it can be used for things like converting angles into vectors, rotating matrices (which I know the barebones of), items that bob up and down in Minecraft or even chiptune music (stuff you would hear on old game consoles like the Gameboy).

That's all from me!