Show that quadratic form $z_0^2+z_1^2+z_2^2$ is invariant under $SO_3(\mathbb{R})$

Question

Let $z=(z_0,z_1,z_2)$. We thus have $z_0^2+z_1^2+z_2^2 = z^Tz=||z||$.

Showing invariant means is this what I need to show:

$$\forall A\in SO_3(\mathbb{R}), \ ||Az||=||z||?$$

But this is clear from definition of $SO_3(\mathbb{R})$: $$||Az||=(Az)^T(Az)=z^TA^TAz=z^TIz=z^Tz=||z||.$$

Am I proving the correct claim here?

Answer 1

Absolutely. The only mistake in your proof is that $z^{T}\cdot z = \|z\|^2$ and not just $\|z\|$. But this doesn't affect the proof.