Question

Show that the magnitude of a vector is unchanged by a rotation of the rectangular coordinate system. Use a counterclockwise rotation about the z axis through an angle θ.

Answer 1

Take a vector from the space,\[\vec{v}=\left[\begin{matrix}a \\ b \\ c\end{matrix}\right]\] with modulus,
\[v=\sqrt{a^2+b^2+c^2}\]and apply the rotation matrix around the z axis,
\[R_z=\left[\begin{matrix} \cos\theta & -\sin\theta & 0\\\sin\theta &\cos\theta & 0 \\ 0 & 0 & 1\end{matrix}\right]\]so,
\[\vec{w}=R\cdot \vec{v}=(a\cos\theta-b\sin\theta,a\sin\theta+b\cos\theta,c)\]
And take the modulus,
\[w^2=(a\cos\theta-b\sin\theta)^2+(a\sin\theta+b\cos\theta)^2+c^2=\]
\[=a^2\cos^2\theta+b^2\sin^2\theta-2ab\sin\theta\cos\theta+\]\[+a^2\sin^2\theta+b^2\cos^2\theta+2ab\sin\theta\cos\theta+c^2=\]
\[=a^2+b^2+c^2\Rightarrow w=v\]

Answer 2

Thanks a bunch! I was stuck at the transformation matrix