Why is $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+2\alpha {{x}_{1}}{{x}_{2}}>0$ for all $({{x}_{1}},{{x}_{2}},{{x}_{3}})\ne (0,0,0)$ if and only if $|\alpha |<1$? Why is $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+2\alpha {{x}_{1}}{{x}_{2}}>0$ for all $({{x}_{1}},{{x}_{2}},{{x}_{3}})\ne (0,0,0)$  if and only if $|\alpha |<1$? I tried solving the inequality and just ended up with  $\alpha >-\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{2{{x}_{1}}{{x}_{2}}}$ (if ${{x}_{1}}{{x}_{2}}>0$ ) and $\alpha <-\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{2{{x}_{1}}{{x}_{2}}}$(if ${{x}_{1}}{{x}_{2}}<0$). 

$2\alpha x_1x_2>-(x_1^2+x_2^2+x_3^2)$
RHS must be smaller than $0$, if LHS>$0$, the equation must work. We only consider LHS<$0$
Then we have,
$-2|\alpha| |x_1||x_2|>-(x_1^2+x_2^2+x_3^2)$
$\Leftarrow2|\alpha| |x_1||x_2|<|x_1|^2+|x_2|^2+|x_3|^2$
$\Leftarrow|\alpha|<\frac{|x_1|^2+|x_2|^2+|x_3|^2}{2 |x_1||x_2|}$
$\frac{|x_1|^2+|x_2|^2+|x_3|^2}{2 |x_1||x_2|}\geqslant1 $ $\Leftarrow$ $|x_1|^2+|x_2|^2+|x_3|^2\geqslant|x_1|^2+|x_2|^2\geqslant2 |x_1||x_2|$$\Leftarrow(|x_1|-|x_2|)^2\geqslant0$
If $|\alpha|>1,$ when $|x_3|=0,|x_1|=|x_2|$,$\frac{|x_1|^2+|x_2|^2+|x_3|^2}{2 |x_1||x_2|}=1<|\alpha| $. Contradict, so $|\alpha|<1$.
Inversely, if $|\alpha|<1,$$\frac{|x_1|^2+|x_2|^2+|x_3|^2}{2 |x_1||x_2|}\geqslant1 > |\alpha|$ 
Q.E.D
