To rotate a vector (x, y) clockwise by an angle θ, you can use the following rotation matrix:
| cos(θ) sin(θ) |
| -sin(θ) cos(θ) |Applying this matrix to the vector (x, y) results in the new vector (x', y'):
x' = x cos(θ) + y sin(θ)
y' = -x sin(θ) + y cos(θ)
Therefore, the rotated vector is (x cos(θ) + y sin(θ), -x sin(θ) + y cos(θ)).
Explanation:
The rotation matrix is derived from trigonometry. It transforms the original vector into a new vector that has been rotated clockwise around the origin (0, 0) by the specified angle θ.
Example:
Let's say you have a vector (1, 0) and you want to rotate it 90 degrees (π/2 radians) clockwise.
- cos(π/2) = 0
- sin(π/2) = 1
So the rotated vector (x', y') becomes:
- x' = 1 0 + 0 1 = 0
- y' = -1 1 + 0 0 = -1
The rotated vector is (0, -1), which is the expected result of rotating (1, 0) 90 degrees clockwise.
Special Case: 90 Degree Clockwise Rotation
A simple case is a 90-degree clockwise rotation. For a vector (x, y), rotating it 90 degrees clockwise results in the vector (y, -x). This avoids the need to calculate trigonometric functions and can be useful in certain applications.
Summary
Rotating a vector (x, y) clockwise by an angle θ is achieved using the formula (x cos(θ) + y sin(θ), -x sin(θ) + y cos(θ)). For a 90-degree clockwise rotation, the simplified formula (y, -x) can be used.
