🔁 Rotation Transformation

Last Updated: Jan 2026

A Rotation Transformation turns a vector or object around the origin (or a fixed point) by a given angle $\theta$, without changing its size or shape.

• Length preserved
• Shape preserved
• Direction changes by angle $\theta$

🗣 Hinglish Tip: Rotation = object ko ghumaana, size same rehta hai

Rotation moves a point/vector anticlockwise or clockwise by an angle $\theta$.

• Positive $\theta$ → Anti-clockwise
• Negative $\theta$ → Clockwise
• Standard rotation is about the origin

Rotation in 2D (About Origin)

1.Rotation Normal Form

$X' = X\cos\theta - Y\sin\theta\\
Y' = X\sin\theta + Y\cos\theta$

2.Rotation Matrix (Anti-clockwise)

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

Transformation Formula

$\vec{v}' = R(\theta)\vec{v}$

Rotation of a Point

Rotate point P(2, 1) by $90^\circ$ anti-clockwise.

Known values

$cos90^\circ = 0,   sin90^\circ = 1$

Matrix

$R(90^\circ) =
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}$

Calculation

Result:

$P'(-1, 2)$

Standard Rotation Angles (Must Remember)

Clockwise Rotation

Clockwise rotation by $\theta$ = Anti-clockwise rotation by $-\theta$

Matrix

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

Example

Rotate (3, 4) by $90^\circ$ clockwise

$(x', y') = (y, -x) = (4, -3)$

Rotation About a Point (h, k)

Steps

1. Translate point to origin
2. Rotate
3. Translate back

Formula

$x' = h + (x - h)\cos\theta - (y - k)\sin\theta \\
y' = k + (x - h)\sin\theta + (y - k)\cos\theta$

Rotation Using Homogeneous Coordinates

Matrix Form

$\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta & \cos\theta & 0 \\
0 & 0 & 1
\end{bmatrix}$

Used in graphics & robotics to combine rotation + translation.

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