🔁 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

Step	Expression	Result
x'	0·2 - 1·1	-1
y'	1·2 + 0·1	2

Result:

 $P'(-1, 2)$

Standard Rotation Angles (Must Remember)

Angle	Result of (x, y)
$90\circ$	(-y, x)
$180\circ$	(-x, -y)
$270\circ$	(y, -x)
$360\circ$	(x, y)

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

Translate point to origin
Rotate
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.

|| राम नाम सत्य है ||

Report An Issue

← Previous Next →