Deriving the Rotation Matrix in 2 Dimensions!

GET 15% OFF EVERYTHING! THIS IS EPIC!

Help me create more free content! =)

Let's continue PFAC! =D Today we ar going to derive the rotation matrix in 2 dimension! I hope you enjoy :)

Visit my website! =)

[arsnakehert]

Mathematician: *struggles to explain what I would need 2D rotations for*
Me, an aspiring game developer and graphics programmer: "this is all just a stepping stone so I can eventually learn quaternions"

[Gameboygenius]

2𝔻 bois are nice and all but I prefer 2𝔻 girls. Not a single dimension more or less.

[scottbretzke]

Interesting that the determinant of the rotational matrix is always unity! Even the 3 dimensional ones!

[adrianrs]

German sense of humor!! thanks for the video!

[JeffreyMarshallMilne]

6:20 "I'm going to make a video on this in the near future" ... *Makes video 325 days later*

[Hobbit]

the column's of the rotation matrix represents the coordinates of the rotated unit vectors i and j.

[anurag]

An alternate derivation using elementary algebra and trigonometry: Let the point to be rotated be (x, y) and the point after rotation of angle q be (a, b). Since distance from origin after rotation must be same:
x²+y²=a²+b²
Distance between the two points:
sqrt((a-x)²+(b-y)²)= (RHS from cosine rule)
ax+by= cosq(x²+y²)
a=(cosq(x²+y²)-by)/x
From a²+b²=x²+y²:
=x²+y²
Solving for b (simplifying above expression into quadratic) we get:

Using quadratic formula and simplifying:
b=cosqx+sinxy
Similary we get,
a=cosqx-sinxy
Which can be represented by matrix multiplication, hence deriving the rotation matrix

[EmissaryOfSmeagol]

When I was learning integral calculus, when we got to integrating with respect to dy, I would always solve the problems by 'rotating' the function and integrate wrt dx. What I really did was integrate what I thought the rotated function would be, so it wasn't rigorous. My prof. told me about this, and this is how I do my dy integrals.

[JanKowalski-zzef]

Maybe you could do a video about rotations in minkowski's spacetime?

[taubone]

Here's a cool trick: multiply out two rotation matrices with angles a and b and you get the formulas for cos(a+b) and sin(a+b). This is how I remember the double angle formulas.

[Mayank-mfxr]

i studied the 2-D rotation matrix for anticlockwise sense . it has (1, 2)th element as +sin(theta) and (2, 1)th element as -sin(theta) . this derivation is actually for a clockwise rotation .

it's the whole system which rotates and rotating the system clockwise has a sense of rotating the position vector anticlockwise, which is exactly what happened in the vid . this would need the inverse of the matrix at hand .

i leveled up today . simply remembered the matrix ( and it's inverse ) in high school . now i need it crucially in physics .

nice video btw papa flammy .

[somealgebraist]

Morpheus is Happy, your doing gods work mon frond

[azimsofi]

wow, this is really exciting thankyou! understood from a-z, well explained.

[ozzyfromspace]

My day was going good but then you rotated it. Now it's going great! ☺️🔥🙌🏽 Keep on being awesome, Papa Flammy. Greetings from the US and Botswana 🎉

[smrtfasizmu]

The other way to do it is to say the unit vectors get rotated by an angle theta. Then the first column of a matrix are the entries where the first unit vector lands (cos theta, sin theta). And the second column of a matrix are entries where the second unit vector lands (cos(pi/2 + theta), sin(pi/2 + theta))
Which is just
(-sin(theta), cos(theta)).

[EmilSilva]

This is exactly what I was searching for!! Thank you so much.

[juliang]

hell yes! This video was awesome I really enjoy this
I will be watching the rest!!!

[mindovermath]

I found you through the suggested videos. Your videos are vey informative! I just made a YouTube Channel due to your inspiration. Keep up the Videos! Just Subbed!

[jardelkaique]

Wonderful video! Thank you for bringing it in a simple way

[Zzznmop]

Ayyy this will be on my final Thursday, I can finally take a break and watch papa flammy rotate mama matrix