Vector transformations are fundamental operations in graphics and geometry. These operations modify the properties of vectors, such as their position, orientation, or scale. In this article, we'll explore some common vector transformations.

Types of Vector Transformations

  1. Translation

    • Shifts a vector to a new position.
    • Formula: ( T(\vec{v}) = \vec{v} + \vec{t} )
    • Example: ( \vec{v} = \begin{bmatrix} 1 \ 2 \end{bmatrix}, \vec{t} = \begin{bmatrix} 3 \ 4 \end{bmatrix} \rightarrow T(\vec{v}) = \begin{bmatrix} 4 \ 6 \end{bmatrix} )

  2. Rotation

    • Rotates a vector around a specific point or axis.
    • Formula: ( R(\vec{v}) = \vec{v} \cdot \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} )
    • Example: ( \vec{v} = \begin{bmatrix} 1 \ 0 \end{bmatrix}, \theta = \frac{\pi}{2} \rightarrow R(\vec{v}) = \begin{bmatrix} 0 \ 1 \end{bmatrix} )

  3. Scaling

    • Changes the length of a vector.
    • Formula: ( S(\vec{v}) = k \vec{v} )
    • Example: ( \vec{v} = \begin{bmatrix} 1 \ 2 \end{bmatrix}, k = 2 \rightarrow S(\vec{v}) = \begin{bmatrix} 2 \ 4 \end{bmatrix} )

  4. Reflection

    • Flips a vector across a specific line or plane.
    • Formula: ( R(\vec{v}) = 2\vec{v} - \vec{v} )
    • Example: ( \vec{v} = \begin{bmatrix} 1 \ 0 \end{bmatrix} \rightarrow R(\vec{v}) = \begin{bmatrix} -1 \ 0 \end{bmatrix} )

Practical Applications

Vector transformations are widely used in various fields, including:

  • Computer Graphics: Transforming objects in 2D and 3D space.
  • Robotics: Controlling the movement and orientation of robots.
  • Computer Vision: Analyzing and processing images and videos.

For more information on vector transformations, you can explore our vector graphics page.

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