🔁 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 θ, without changing its size or shape.

  • Length preserved
  • Shape preserved
  • Direction changes by angle θ

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

Rotation moves a point/vector anticlockwise or clockwise by an angle θ.

  • Positive θ → Anti-clockwise
  • Negative θ → Clockwise
  • Standard rotation is about the origin

Rotation in 2D (About Origin)

1.Rotation Normal Form

X′ = Xcosθ − Ysinθ
Y′ = Xsinθ + Ycosθ

2.Rotation Matrix (Anti-clockwise)

R(θ) =
[ cosθ   −sinθ
  sinθ    cosθ ]

Transformation Formula

v⃗′ = R(θ) v⃗

Rotation of a Point

Rotate point P(2, 1) by 90° anti-clockwise.

Known values

cos 90° = 0,   sin 90° = 1

Matrix

R(90°) =
[ 0  −1
  1   0 ]

Calculation

StepExpressionResult
x′0·2 − 1·1−1
y′1·2 + 0·12

Result:

P'(-1, 2)

Standard Rotation Angles (Must Remember)

AngleResult of (x, y)
90°(−y, x)
180°(−x, −y)
270°(y, −x)
360°(x, y)

Clockwise Rotation

Clockwise rotation by θ = Anti-clockwise rotation by −θ

Matrix

R(−θ) =
[ cosθ   sinθ
 −sinθ  cosθ ]

Example

Rotate (3, 4) by 90° 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)cosθ − (y − k)sinθ
y′ = k + (x − h)sinθ + (y − k)cosθ

Rotation Using Homogeneous Coordinates

Matrix Form

[ cosθ  −sinθ   0
  sinθ   cosθ   0
   0      0     1 ]

Used in graphics & robotics to combine rotation + translation.

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