Question

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

$\Rightarrow \left[\begin{array}{ccc}a& b& c\end{array}\right]=0$

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

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

$\Rightarrow -\beta +1=0\Rightarrow \beta =1$

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

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

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

$\Rightarrow \alpha =\pm 1$