Question

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

Answer 1

As a, b, c are linearly dependent

$\left[abc\right]=0$

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

$C_3\to C_3-C_1$

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

Expanding determinant w.r.t $C_3$

$(\beta -1)(3-4)=0$

$-1(\beta -1)=0\Rightarrow \beta =1$

Given $|C|=\sqrt{3}$

$c=\hat{i}+\alpha \hat{j}+\beta \hat{k}$

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

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

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

${\alpha }^2=1\alpha =\pm 1$