Let \(\vec{A} = 2\hat{i} + \hat{j} – 2\hat{k}\) and \(\vec{B} = \hat{i} + \hat{j}\), If \(\vec{C}\) is a vector such that \(|\vec{C} – \vec{A}| = 3\) and the angle between A × B and C is \({30^{\circ}}\), then \(|(\vec{A} × \vec{B}) × \vec{C}|\) = 3 then the value of \(\vec{A}.\vec{C}\) is equal to
\(\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -2 \\ 1 & 1 & 0 \end{vmatrix} = 2\hat{i} - 2\hat{j} + \hat{k}\), so \(|\vec{A} \times \vec{B}| = 3\).
\(|(\vec{A} \times \vec{B}) \times \vec{C}| = |\vec{A} \times \vec{B}||\vec{C}|\sin 30° = \frac{3|\vec{C}|}{2} = 3 \Rightarrow |\vec{C}| = 2\)
\(|\vec{C} - \vec{A}|^2 = |\vec{C}|^2 + |\vec{A}|^2 - 2\vec{A} \cdot \vec{C}\): \(9 = 4 + 9 - 2\vec{A} \cdot \vec{C} \Rightarrow \vec{A} \cdot \vec{C} = 2\)
Answer: (B) 2
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