Rotating a Parabola: 30o Anti-Clockwise About Origin

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SUMMARY

The discussion focuses on the mathematical transformation of the parabola defined by the equation y = x² when rotated anti-clockwise by 30 degrees around the origin. The new equation of the curve can be derived using the rotation matrix, which is given by R(θ) = \begin{bmatrix}cos(30°) & -sin(30°) \\ sin(30°) & cos(30°)\end{bmatrix}. By applying this matrix to a point (x, x²), the coordinates transform accordingly, resulting in a new equation that represents the rotated parabola.

PREREQUISITES
  • Understanding of coordinate transformations
  • Familiarity with rotation matrices
  • Knowledge of trigonometric functions (sine and cosine)
  • Basic algebra for manipulating equations
NEXT STEPS
  • Study the derivation of rotation matrices in 2D geometry
  • Learn how to apply transformations to conic sections
  • Explore the implications of rotating curves in calculus
  • Investigate the effects of different angles on the shape of parabolas
USEFUL FOR

Mathematicians, physics students, and educators interested in geometric transformations and conic sections will benefit from this discussion.

squenshl
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The parabola y = x2 is rotated anti-clockwise about the origin through the angle 30o. What is the new equation of the new curve (relative to the standard basis)?
 
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If you have a point (x,x2) and you rotate it by 30 degrees around the origin what does it become?
 
And you can rotate a point (x,y) through an angle [itex]\theta[/itex] about the origin by the matrix multiplication
[tex]\begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}[/tex].
 

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