ok, some context.

i have this yearly ritual of rewatching 3b1b’s Essence of Linear Algebra, hoping that this time i’ll finally understand what everything’s about. i’ve been doing this since 9th grade. this is almost year five. time flies lol.

but until this year, i never bothered to ask myself the million-dollar question:

where the fuck does this rotation matrix come from?

mind you, i’ve copied its implementation into my codebases more times than i can count, across multiple projects, multiple languages. you just plug in some angles (alpha, beta, gamma, whatever the fuck), multiply a vector, and boom — it rotates???

wym bruh???

this state of ignorance was about to get checked.

during one of my daily youtube doomscroll sessions, i came across a video by tsoding where he was demonstrating why graphics APIs are largely irrelevant if you actually understand the math and what you’re trying to achieve. somewhere in the video, he casually implements a 3D rotation matrix in C (i don’t remember tho).

he’s someone i really look up to. his streams are pure gold if you’re even slightly interested in low-level programming or graphics. when he admitted that he too took rotation matrices for granted for a long time, i felt violently seen.

but then he mentioned a video where someone derives the rotation matrix.

finally. it was time.

i watched it and,

OH MY FUCKING GOD.

i already knew this shit.

not vaguely. not approximately. i knew it exactly. and i even remember where i learned it from.

let me show you.

what’s a rotation matrix?

let’s stay in 2D for now.

say you have a point \( P = (1, 0) \). it lies on the x-axis. if you rotate it 90° counterclockwise, it ends up at \( (0, 1) \).

a rotation matrix is just a compact way to store the information needed to perform this rotation.

you take the position vector of the point, multiply it by the matrix, and out comes the rotated vector.

no magic. just simple vector-matrix multiplication.

can i use this for any angle?

yep.

where does the matrix actually come from?

alright, let’s cook.

say we have a vector \( \vec{v} \):

$$ \vec{v} = \begin{bmatrix} x \\ y \end{bmatrix} $$

using basic trigonometry, we can describe this vector by it’s magnitude and angle from the x-axis:

$$ \begin{aligned} x &= |\vec{v}|\cos\alpha \\ y &= |\vec{v}|\sin\alpha \end{aligned} $$

where \( \alpha \) is the angle the vector makes with the x-axis.
(unit circle shit. you know this.)

now, one important thing about rotation:

rotation does not change the magnitude of a vector.
it only changes its direction.

so if we rotate this vector by an angle \( \beta \), the new components become:

$$ \begin{aligned} x' &= |\vec{v}|\cos(\alpha + \beta) \\ y' &= |\vec{v}|\sin(\alpha + \beta) \end{aligned} $$

and yes, this is where those ancient high-school identities crawl out of the grave:

$$ \begin{aligned} \cos(\alpha + \beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ \sin(\alpha + \beta) &= \cos\alpha\sin\beta + \sin\alpha\cos\beta \end{aligned} $$

plugging these in:

$$ \begin{aligned} x' &= |\vec{v}|(\cos\alpha\cos\beta - \sin\alpha\sin\beta) \\ y' &= |\vec{v}|(\cos\alpha\sin\beta + \sin\alpha\cos\beta) \end{aligned} $$

now here’s the part where everything clicks.

we already know:

$$ |\vec{v}|\cos\alpha = x \quad \text{and} \quad |\vec{v}|\sin\alpha = y $$

so substitute:

$$ \begin{aligned} x' &= x\cos\beta - y\sin\beta \\ y' &= x\sin\beta + y\cos\beta \end{aligned} $$

and now we write that as a matrix multiplication:

$$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos\beta & -\sin\beta \\ \sin\beta & \cos\beta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

that’s it.

that’s the 2D rotation matrix:

$$ R(\beta) = \begin{bmatrix} \cos\beta & -\sin\beta \\ \sin\beta & \cos\beta \end{bmatrix} $$

so… what’s actually inside the matrix?

here’s the part no one explicitly tells you:

a matrix stores what happens to the basis vectors.

in 2D, your basis vectors are:

  • \( (1, 0) \) is the x-axis
  • \( (0, 1) \) is the y-axis

rotate \( (1, 0) \) by \( \beta \):
→ \( (\cos\beta, \sin\beta) \)

rotate \( (0, 1) \) by \( \beta \):
→ \( (-\sin\beta, \cos\beta) \)

those two results become the columns of the matrix.

that’s it.

so when you multiply a vector by the rotation matrix, you’re just expressing that vector in terms of rotated axes.

it’s honestly wild how much stuff we learn in school that feels useless, until one day you see it applied somewhere real, and suddenly it clicks.

CLOCK THAT SHIT 🤏.

PS: here’s the link to the video i watched in which the guy derives it soo good, uhhhhhhhhhhhhhhhhhhhhhhhhhh!!!!