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

$$\begin{gathered} Given vectors are \\ \overline{a}=\overline{i}+\overline{j}+\overline{k},\overline{ b}=4\overline{i}+3\overline{j}+4\overline{k},\overline{c}=\overline{i}+\alpha \overline{j}+\beta \overline{k} \\ \sin ce given vectors are linearly dependent then \\ \left|\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\alpha \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\beta \end{array}\right|=0\Rightarrow 1(3\beta -4\alpha )-1(4\beta -4)+1(4\alpha -3)=0 \end{gathered}$$

$\Rightarrow 3\beta -4\alpha -4\beta +4+4\alpha -3=0\Rightarrow -\beta =-1\Rightarrow \beta =1$

 $$\begin{gathered} Given \left|\overline{c}\right|=\sqrt{3}\Rightarrow 1+{\alpha }^2+{\beta }^2=3\Rightarrow {\alpha }^2+{\beta }^2=2 \\ \Rightarrow {\alpha }^2+1=2 \\ \Rightarrow {\alpha }^2=1 \\ \Rightarrow \alpha =\pm 1 \end{gathered}$$

$\therefore \alpha =\pm 1,\beta =1$