If $\overline{a}=\overline{i}+\overline{j}+\overline{k}$, $\overline{\mathrm{b}}=4\overline{i}+3\overline{j}+4\overline{k}$, $\overline{\mathrm{c}}=\overline{i}+\mathrm{\alpha }\overline{\mathrm{j}}+\mathrm{\beta }\overline{\mathrm{k}}$  be linearly dependent vectors and $\left|\overline{\mathrm{c}}\right|=\sqrt{3}$ then 

Vectors are linearly dependent
$\begin{array}{l}\Rightarrow \left|\begin{array}{lll}1& 1& 1\\ 4& 3& 4\\ 1& \mathrm{\alpha }& \mathrm{\beta }\end{array}\right|=0\Rightarrow 1(3\mathrm{\beta }-4\mathrm{\alpha })-1(4\mathrm{\beta }-4)+1(4\mathrm{\alpha }-3)=0\Rightarrow \mathrm{\beta }=1,{\left.\overline{\mathrm{c}}|=\sqrt{3}\Rightarrow |\overline{\mathrm{c}}\right|}^2=3\\ \Rightarrow 1+{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2=3\Rightarrow {\mathrm{\alpha }}^2=1\Rightarrow \mathrm{\alpha }=\pm 1\end{array}$