Question

If $a=i+j+k,b=4i+3j+4k$ and $c=i+\alpha j+\beta k$ are linearly dependent vectors and $|c|=\sqrt{3}$, then the values of $\alpha$ and $\beta$ are respectively.

• A. $\pm 1,1$
• B. $\pm 2,1$
• C. $0,\pm 1$
• D. None of these

Answer 1

The correct option is A $\pm 1,1$

$\because$ $a,b$ and $c$ are linearly dependent vectors
$\Rightarrow \left[abc\right]=0$

$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 4& 3& 4\\ 1& \alpha & \beta \end{array}\right|$
$\Rightarrow 1(3\beta -4\alpha )-1(4\beta -4)+1(4\alpha -3)=0$
$\Rightarrow -\beta +1=0\Rightarrow \beta =1$

Now, $|c|=\sqrt{3}$

$\Rightarrow \sqrt{1+{\alpha }^2+{\beta }^2}=\sqrt{3}$
$\Rightarrow 1+1+{\alpha }^2=3\Rightarrow {\alpha }^2=1$

$\therefore \alpha =\pm 1$