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How do you rotate a vector counterclockwise?

Published in Vector Rotation 2 mins read

You can rotate a vector counterclockwise by rotating it around a coordinate axis. To visualize counterclockwise rotation, imagine looking straight at the axis of rotation towards the origin.

Rotating a Vector in 3D Space

A fundamental way to rotate a vector in three dimensions is around one of the three coordinate axes: the x-axis, y-axis, or z-axis.

Rotations Around Specific Axes

  • Rotation around the x-axis: Imagine looking down the x-axis toward the origin. A counterclockwise rotation would then move the vector in a circular path going from the positive y-axis toward the positive z-axis, then to the negative y-axis, and finally to the negative z-axis.

  • Rotation around the y-axis: Envision looking down the y-axis toward the origin. A counterclockwise rotation around the y-axis would make the vector move from the positive z-axis toward the positive x-axis, then to the negative z-axis, and finally to the negative x-axis.

  • Rotation around the z-axis: Imagine looking down the z-axis toward the origin. A counterclockwise rotation would make the vector move from the positive x-axis toward the positive y-axis, then to the negative x-axis, and finally to the negative y-axis.

Using Rotation Matrices

Mathematically, rotations are often achieved using rotation matrices. Each axis has a corresponding rotation matrix, and when multiplied with the vector, it results in the rotated vector.


Rotation Axis Rotation Matrix
Around the x-axis 
  [ 1   0    0  ]
  [ 0  cosθ -sinθ]
  [ 0  sinθ  cosθ]

|
| Around the y-axis |

  [ cosθ   0  sinθ]
  [  0    1   0  ]
  [-sinθ   0  cosθ]

|
| Around the z-axis |

 [cosθ -sinθ   0]
  [sinθ  cosθ   0]
  [  0    0   1]

|


  • θ (theta) represents the angle of rotation in radians.


Practical Insight: Understanding the counterclockwise rotation around each axis can be crucial in various applications, such as:

  • Computer Graphics: Animating objects and manipulating viewpoints.
  • Robotics: Controlling robot arm movements and orientation.
  • Physics and Engineering: Modeling and simulation of physical systems.
  • Navigation: Calculating changes in orientation.

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