For three vectors $\overrightarrow{A}=2\hat{j}-3\hat{k},\overrightarrow{B}=-2\hat{i}+\hat{k}and\hat{C}=3\hat{i}-\hat{j},$ where î, ĵ and k̂ are unit vectors along the axes of a right-handed rectangular/Cartesian coordinate system, the value of $\left(\overrightarrow{A.}\left(\overrightarrow{B}\times \overrightarrow{C}\right)+6\right)$ is

Answer :

Correct answer is : 6

$\overrightarrow{A}=2\hat{j}-3\hat{k},\overrightarrow{B}=-2\hat{i}+\hat{k}and\hat{C}=3\hat{i}-\hat{j}$

⇒ $\overrightarrow{\mathrm{A}.}\left(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{C}}\right)=\left|\begin{array}{ccc}0& 2& -3\\ -2& 0& 1\\ 3& -1& 0\end{array}\right|$

⇒ $\overrightarrow{A}\left(\overrightarrow{B}\times \overrightarrow{C}\right)=\left(-2\right)\left(-3\right)-3\left(2\right)$

⇒ $\overrightarrow{\mathrm{A}.}\left(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{C}}\right)=0$

∴ $\overrightarrow{\mathrm{A}.}\left(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{C}}\right)+6=6$