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A rotation matrix is a type of linear transformation that rotates points in the plane around the origin. In simple terms, it shifts the position of a point but keeps its distance from the origin constant.
To understand better, imagine spinning a piece of paper on a table around a pin stuck in the middle. Mathematically, we describe this rotation using a rotation matrix.
For any angle \(\theta\), the rotation matrix in \(\mathbb{R}^{2}\) (2-dimensional space) is represented as:
\[ R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} \]
When we multiply this matrix by any vector, it rotates the vector counterclockwise by the angle \(\theta\). This is useful in graphics, physics, and engineering, where rotation of objects is a common task.

At the heart of a rotation matrix are the trigonometric functions cosine (\(\cos\)) and sine (\(\sin\)). These functions relate the angles of a triangle to the lengths of its sides and are crucial in defining how much a point in \(\mathbb{R}^{2}\) gets rotated.
For the given problem, we use the specific angle \(\theta = \frac{2 \pi}{3}\). Knowing the exact trigonometric values makes it easier to compute the rotation matrix. Here you need:

\[- \cos\left( \frac{2 \pi}{3} \right) = -\frac{1}{2}\] \[- \sin\left( \frac{2 \pi}{3} \right) = \frac{\sqrt{3}}{2}\]

Substitute these into the rotation matrix to get the specific matrix for rotating points through an angle of \(\frac{2 \pi}{3}\).
By inserting these values, our matrix becomes:
\[ R\left( \frac{2 \pi}{3} \right) = \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix} \]
This matrix will rotate any point in \(\mathbb{R}^{2}\) by 120 degrees counterclockwise.

In mathematics, \(\mathbb{R}^{2}\), or ¸é², signifies the 2-dimensional Euclidean space. This is simply the plane we are familiar with, where any point can be represented by two coordinates (x, y).
Linear transformations in \(\mathbb{R}^{2}\) include operations like rotations, translations, and scaling of points in this plane.
When working with rotations in \(\mathbb{R}^{2}\), the points on this plane are altered. However, their fundamental relationships in terms of distance and angles remain intact. For example:

• A point (1, 0) in \(\mathbb{R}^{2}\) rotated by \(\frac{2 \pi}{3}\) will land on a new position in \(\mathbb{R}^{2}\), becoming approximately (-0.5, \(\frac{\sqrt{3}}{2}\)).
• This means the original position shifted, but its length from the origin remained constant.

Understanding the behavior of transformations in \(\mathbb{R}^{2}\) helps us manipulate shapes and points effectively, crucial in fields like computer graphics, navigation, and more.