Vector Magnitude is Invariant Under Rotation/Proof 2
Jump to navigation
Jump to search
Theorem
Let $\mathbf v$ be an arbitrary vector in the Cartesian plane $\CC$.
Let the coordinate system then be rotated in the anticlockwise direction by an arbitrary angle $\theta$.
Then:
the magnitude of $\mathbf v$ is unchanged in the new coordinate system.
Proof
We offer three equivalent statements:
By definition, rotation of the coordinate system affects the coordinates and not the vector.
This article needs to be tidied. Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Tidy}} from the code. |
Rotation is a rigid transformation. It does not change side lengths or angles.
The equations of rotation of coordinates are linear transformations.
$\blacksquare$