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Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In the context of vector rotation, trigonometry plays a vital role because it defines how angles can be used to change the direction of a vector in a two-dimensional space. The key functions used in trigonometry for rotations are the sine and cosine functions.
These functions help describe how much a vector is shifted along each axis when rotated. For example, when a vector is rotated by an angle \( \theta \), the new position of the vector depends on these trigonometric functions:

• \( \cos(\theta) \) determines the scaling factor in the direction of the x-axis.
• \( \sin(\theta) \) determines the scaling factor in the direction perpendicular to the x-axis.

In simple words, trigonometry allows us to understand how rotating an object by a certain angle will affect its position in space.

Linear algebra is a field of mathematics focused on vector spaces and linear mappings between these spaces. It is crucial for understanding and performing vector rotations using matrices. A matrix is a rectangular array of numbers arranged in rows and columns, which can represent linear transformations such as rotations, translations, and scaling in multi-dimensional space.
When we talk about rotating vectors, we use a specific type of matrix known as the rotation matrix. This matrix allows us to change the angle and direction of a vector while preserving its magnitude, achieving a new position in a systematic and calculable way. A 2D rotation matrix for rotating a vector by an angle \( \theta \) is defined as:

• For a clockwise rotation, the matrix is:
  \[ \begin{bmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{bmatrix} \]

By multiplying this matrix with a vector, we effectively rotate the vector in the 2D space by the desired angle.

Vector rotation involves changing the direction of a vector by a specified angle using a rotation matrix. This is a common operation in many fields, including physics, computer graphics, and engineering.
To rotate a vector, we take the following steps:

• Identify the vector to be rotated. In our example, the vector is \( \begin{bmatrix} 5 \ -3 \end{bmatrix} \).
• Decide the angle of rotation, which is \( \theta = \frac{\pi}{7} \) in our case.
• Apply the rotation matrix to the vector, utilizing trigonometric functions to obtain the new coordinates.

Doing these steps allows us to achieve the rotated vector coordinates, demonstrating how the direction and orientation of the original vector changes while its origin point remains fixed. The final result is a new vector, expressing the rotated position in terms of its components along the x and y axes.