Area of the parallelogram, whose adjacent sides are given by the vectors \(\vec{a} = 2\hat{\imath} - \hat{\jmath} + 5\hat{k}\) and \(\vec{b} = 2\hat{\imath} + \hat{\jmath} + 2\hat{k}\), is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\sqrt{101}\)
The area of a parallelogram with adjacent sides \(\vec{a}\) and \(\vec{b}\) is \(|\vec{a}\times\vec{b}|\), the magnitude of the cross product:
\( \displaystyle \vec{a}\times\vec{b} = \begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\2 & -1 & 5\\2 & 1 & 2\end{vmatrix} \)
\(= (-2 - 5)\hat{\imath} - (4 - 10)\hat{\jmath} + (2 + 2)\hat{k}\)
\(= -7\hat{\imath} + 6\hat{\jmath} + 4\hat{k}\)
\( \displaystyle |\vec{a}\times\vec{b}| = \sqrt{49 + 36 + 16} = \sqrt{101} \)