Q72 / 150 · MHT CET 2026 (20 April Shift 2) · Mathematics
The volume of the parallelopiped whose coterminous edges are $2\hat{i} + \hat{j} - \hat{k}, 3\hat{i} - \hat{j} - \hat{k}, \hat{j} + 3\hat{k}$ is
- Correct answerA$16$ cu. units
- B$6$ cu. units
- C$2$ cu. units
- D$12$ cu. units
Solution
Let the coterminous edges be $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{b} = 3\hat{i} - \hat{j} - \hat{k}$, and $\vec{c} = \hat{j} + 3\hat{k}$.
The volume of the parallelopiped is given by the absolute value of the scalar triple product $[\vec{a} \ \vec{b} \ \vec{c}]$.
$[\vec{a} \ \vec{b} \ \vec{c}] = \begin{vmatrix} 2 & 1 & -1 \\ 3 & -1 & -1 \\ 0 & 1 & 3 \end{vmatrix}$
Expanding the determinant along the first row:
$[\vec{a} \ \vec{b} \ \vec{c}] = 2(-3 + 1) - 1(9 - 0) - 1(3 - 0)$
$[\vec{a} \ \vec{b} \ \vec{c}] = -4 - 9 - 3 = -16$
$V = |-16| = 16$ cu. units.
Answer: $16$ cu. units
The volume of the parallelopiped is given by the absolute value of the scalar triple product $[\vec{a} \ \vec{b} \ \vec{c}]$.
$[\vec{a} \ \vec{b} \ \vec{c}] = \begin{vmatrix} 2 & 1 & -1 \\ 3 & -1 & -1 \\ 0 & 1 & 3 \end{vmatrix}$
Expanding the determinant along the first row:
$[\vec{a} \ \vec{b} \ \vec{c}] = 2(-3 + 1) - 1(9 - 0) - 1(3 - 0)$
$[\vec{a} \ \vec{b} \ \vec{c}] = -4 - 9 - 3 = -16$
$V = |-16| = 16$ cu. units.
Answer: $16$ cu. units
Hence, the correct answer is option: 'A'.