Prove that a specific map induces an embedding from $\mathbb{R}P^{2}$ into $\mathbb{R}^6$

Question

Can someone help me:

The map $f : {S}^2 \rightarrow \mathbb{R}^6$ given by $f (x; y; z) = (x^2, y^2, z^2, \sqrt{2}yz, \sqrt{2}xz, \sqrt{2}xy)$ induces an embedding $C^{\infty}$ of $\mathbb{R}P^2$ in $\mathbb{R}^6$?

How can we get from $f$ a embedding $C^{\infty}$ of $\mathbb{R}P^2$ in $\mathbb{R}^4$?