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

$$\begin{gathered} Given vectors a=\overline{i}+\overline{j}+\overline{k},b=4\overline{i}+3\overline{j}+4\overline{k},c=\overline{i}+\alpha \overline{j}+\beta \overline{k} \\ \sin ce given vectors are lineaarly dependent then \\ \left|\begin{array}{ccc}1& 1& 1\\ 4& 3& 4\\ 1& \mathrm{\alpha }& \mathrm{\beta }\end{array}\right|=0 \\ \Rightarrow 1(3\beta -4\alpha )-1(4\beta -4)+1(4\alpha -3)=0 \\ \Rightarrow \beta =1 \\ Given \left|\overline{c}\right|=\sqrt{3} \\ \Rightarrow 1+{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2=3\Rightarrow \mathrm{\alpha }=\pm 1 \\ \therefore \mathrm{\alpha }=\pm 1,\mathrm{\beta }=1 \end{gathered}$$