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
Answer & explanation
Correct answer: option 4
$\vec a,\vec b,\vec c$ 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,
$|\vec c|=\sqrt{3}$
$⇒\sqrt{1+α^2+β^2}=\sqrt{3}$
$⇒1+1+α^2=3⇒α^2=1⇒α=±1$