91Ó°ÊÓ

The concept of a rotation matrix is a fundamental idea in transforming coordinates. A rotation matrix allows you to rotate points in a plane without changing their relative distances. It effectively alters the direction in which points are viewed, enabling a new perspective on the plane.

The rotation matrix for an angle \( \theta \) in a two-dimensional coordinate system is expressed as:\[ R = \begin{bmatrix}\cos(\theta) & -\sin(\theta) \sin(\theta) & \cos(\theta)\end{bmatrix} \]This matrix is special because it preserves lengths and angles, making it an orthogonal matrix. When you multiply a point's coordinates by this matrix, the point rotates counterclockwise by the angle \( \theta \).

In the given exercise, you're rotating a point by \( 60^\circ \). Calculating \( \cos(60^\circ) = \frac{1}{2} \) and \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), the rotation matrix becomes:\[ R = \begin{bmatrix}\frac{1}{2} & -\frac{\sqrt{3}}{2} \frac{\sqrt{3}}{2} & \frac{1}{2}\end{bmatrix} \] This matrix can then be used to transform any coordinate from the old system to the new rotated system.

Rotating a coordinate system is a creative way to view geometric objects under a different light. By rotating the entire system, both the axes and the points transform, leading to new coordinate values and relationships.

For any point \((x, y)\), you find its rotated coordinates \((x', y')\) by applying the rotation matrix derived from the chosen angle \( \theta \). This transformation changes \((x, y)\) using:\[\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix}\cos(\theta) & -\sin(\theta) \sin(\theta) & \cos(\theta)\end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}\]
In practical terms, you rotate the axes themselves while maintaining the structure and relationships of the object in space. With our exercise, starting with coordinates \((-2, 6)\), the transformation through the 60° rotation matrix results in new coordinates \((x', y') = \left(-1 - 3\sqrt{3}, 3 - \sqrt{3}\right)\).

This new perspective helps to understand the positioning and orientation of geometric forms in different contexts.

Transforming equations when rotating a coordinate system can appear challenging, but it's just an extension of coordinate changes. You replace the original variables using the relationships defined by the rotation matrix.

To transform an equation, you need to express the original \((x, y)\) in terms of the rotated coordinates \((x', y')\). As shown earlier for the specific rotation:

• \(x = \frac{1}{2}x' - \frac{\sqrt{3}}{2}y'\)
• \(y = \frac{\sqrt{3}}{2}x' + \frac{1}{2}y'\)

Substitute these into the original equation to obtain the equivalent expression.

For the exercise, transforming \( \sqrt{3}xy + y^2 = 6 \) involves substituting the above expressions for \(x\) and \(y\) into the equation. Simplifying this replaced form yields the equation in the \((x', y')\) space.

This process unveils how shifts in perspective (here, through rotation) alter the mathematical representation of curves, giving insight into their intrinsic geometric properties.