Question

Show that the magnitude of a vector is unchanged by a rotation. hint using the matrix [cos∅ -sin∅ 0 ; sin∅ cos∅ 0 ; 0 0 1] for a rotation about the z-axis through an an angle ∅.

Answer 1

Take any vector in
`\mathbb R^3` and apply the rotation transformation.


`\begin{bmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}x\cos\theta-y\sin\theta\\x\sin\theta+y\cos\theta\\z\end{bmatrix}`

Take the norm of the transformed vector.thet


`√((x\cos\theta-y\sin\theta)^2+(x\sin\theta+y\cos\theta)^2+z^2)`

`=√(x^2\cos^2\theta-2xy\cos\theta\sin\theta+y^2\sin^2\theta+x^2\sin^2\theta+2xy\sin\theta\cos\theta+y^2\cos^2\theta+z^2)`

`=√(x^2(\cos^2\theta+\sin^2\theta)+y^2(\sin^2\theta+\cos^2\theta)+z^2)`

`=√(x^2+y^2+z^2)`

which is exactly the norm of
`\begin{bmatrix}x\\y\\z\end{bmatrix}`.