If $\vec a =\hat i+\hat j+\hat k, \vec b =4\hat i+3\hat j+4\hat k$ and $\vec c=\hat i+α\hat j+β\hat k$ are linearly dependent vectors and $|\vec c|=\sqrt{3}$, then

$α =±1, β=1$

$\vec a,\vec b,\vec c$ are linearly dependent vectors.

$∴[\vec a\,\,\vec b\,\,\vec c]=0$

$⇒\begin{vmatrix}1&1&1\\4&3&4\\1&α&β\end{vmatrix}=0⇒-β+1=0⇒β=1$

Also,

$|\vec c|=\sqrt{3}$

$⇒\sqrt{1+α^2+β^2}=\sqrt{3}$

$⇒1+1+α^2=3⇒α^2=1⇒α=±1$