Let $\mathbf{x}$ and $\mathbf{y}$ be non-zero vectors in a real vector space with scalar product denoted by $\mathbf{x} \cdot \mathbf{y}$. Prove that $(\mathbf{x} \cdot \mathbf{y})^{2} \leqslant(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y})$, and prove also that $(\mathbf{x} \cdot \mathbf{y})^{\mathbf{2}}=(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y})$ if and only if $\mathbf{x}=\lambda \mathbf{y}$ for some scalar $\lambda$.

(i) By considering suitable vectors in $\mathbb{R}^{3}$, or otherwise, prove that the inequality $x^{2}+y^{2}+z^{2} \geqslant y z+z x+x y$ holds for any real numbers $x, y$ and $z$.

(ii) By considering suitable vectors in $\mathbb{R}^{4}$, or otherwise, show that only one choice of real numbers $x, y, z$ satisfies $3\left(x^{2}+y^{2}+z^{2}+4\right)-2(y z+z x+x y)-4(x+y+z)=0$, and find these numbers.