The area of the parallelogram whose adjacent sides are determined by the vectors \(\vec{a} = \hat{\imath} - \hat{\jmath} + 3\hat{k}\) and \(\vec{b} = 2\hat{\imath} - 7\hat{\jmath} + \hat{k}\) is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(15\sqrt{2}\)
The area of a parallelogram with adjacent sides \(\vec{a}\) and \(\vec{b}\) is \(|\vec{a} \times \vec{b}|\).
\( \displaystyle \vec{a} \times \vec{b} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ 1 & -1 & 3 \\ 2 & -7 & 1 \end{vmatrix} \)
\( = (-1 + 21)\hat{\imath} - (1 - 6)\hat{\jmath} + (-7 + 2)\hat{k} \)
\( = 20\hat{\imath} + 5\hat{\jmath} - 5\hat{k} \)
So the area is
\( \displaystyle |\vec{a} \times \vec{b}| = \sqrt{400 + 25 + 25} = \sqrt{450} \)
\( = 15\sqrt{2} \)
NCERT (2026-27 edition):
Vector (or Cross) Product of Two Vectors
