Vector Magnitude is Invariant Under Rotation/Proof 2

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.

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