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Analytic Geometry2D

Translation and Rotation of Axes - Theory & Concepts - Analytic Geometry Axes Rotation Matrix Explorer

Transformation of coordinates through translation and rotation of the axes, and eliminating the xy-term in conic sections.

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Rotation of Axes Matrix Explorer

Rotation Control

Rotation Angle θ30°

Draggable point coordinates

Original Coordinates (x, y)
(5.0, 4.0)
Rotated Coordinates (x', y')
(6.33, 0.96)

Transformation Matrix solver

Rotation Transformation Matrix
(x′y′)=(cos⁡θsin⁡θ−sin⁡θcos⁡θ)(xy)\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}
Step 1: Trigonometric Values
θ=30∘  ⟹  cos⁡θ=0.866, sin⁡θ=0.500\theta = 30^{\circ} \implies \cos\theta = 0.866, \ \sin\theta = 0.500
Step 2: Linear Substitution
x′=5⋅(0.866)+4⋅(0.500)=6.33x' = 5 \cdot (0.866) + 4 \cdot (0.500) = 6.33
y′=−5⋅(0.500)+4⋅(0.866)=0.96y' = -5 \cdot (0.500) + 4 \cdot (0.866) = 0.96
xyx'y'Point P (5.0, 4.0)P(5.0, 4.0)

Drag point P or adjust angle θ to see original vs. rotated projection lines

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