Derive the rotation matrix in 30 seconds

 

 

Hmm… Annoying…

 

What’s wrong with you?

 

Ah, Mr. Penguin. I’d like to understand why the rotation matrix is expressed in the figure below but…

 

 

All website I’ve checked have so many calculations that I’m getting tired of it..

 

Rally? Actually, you can derive the rotation matrix in 30 seconds.

 

Seriously? Would you tell me the way?

 

Okay. Firstly, please tell me the coordinates of the two points in the figure below.

 

 

That’s easy. Here you are.

 

 

Good. Actually, there is another way to describe the points by using the rotation matrix R.

Please take a look at the figure below.

 

 

We haven’t known the values of the rotation matrix R are but yeah, if we can use the rotation matrix R, the points can be described like that.

 

We are going to go to the next step.

In the right side of the figure below, I made a point [x,y] rotate by Φ. You don’t have any questions regarding the rotated coordinate, right?

 

 

Yeah, that’s what the rotation matrix means. (I haven’t known the values of the rotation matrix R though…)

 

I added a equation in the right bottom side of the figure below. I just expanded the coordinate of the rotated point.

You don’t have any questions either, right?

 

 

No questions. You just expanded R[x,y].

 

By the way, did you know that we’ve already known the values of R[1,0] and R[0,1]?

 

Eh… Ah, you are right. We’ve already got them.. The values are inside the red boxes.

 

 

Will you substitute the values into the equation in the right bottom side in the figure above?

 

Sure.. Is this right?

 

 

Did you realize something?

 

Well… Nope..

 

Really? Will you pay attention to the equation you derived more carefully?

 

 

Ah, we’ve got the values of the rotation matrix…

 

Then, see you again.

 

Thank you so much.