Question

Let $\alpha =2\hat{i}+3\hat{j}-\hat{k}$ and $\overline{\beta }=\hat{i}+\hat{j}$. if $\gamma$ is unit vector, then the maximum value of $[\overline{\alpha }\times \overline{\beta }\overline{\beta }\times \gamma \overline{\gamma }\times \overline{\alpha }]$is equal to

• A. $1$
• B. $3$
• C. $4$
• D. $9$

Answer 1

The correct option is A $1$

$\alpha \times \beta =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& -1\\ 1& 1& 0\end{array}\right|$

$=(0+1)\hat{i}-(0+1)\hat{j}+(2-3)\hat{k}$

$=\hat{i}-\hat{j}-\hat{k}$

$\beta \times \gamma =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 0\\ 1& 1& 1\end{array}\right|$

$=(1-0)\hat{i}-(1-0)\hat{j}+(1-1)\hat{k}$

$=\hat{i}-\hat{j}+0\hat{k}$

$\gamma \times \alpha =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 2& 3& -1\end{array}\right|$

$=(-1-3)\hat{i}-(-1-2)\hat{j}+(3-2)\hat{k}$

$=-4\hat{i}+3\hat{j}+\hat{k}$

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

$=1(-1-0)+1(1-0)-1(3-4)$

$=-1+1+1=1$