Question

The volume of the parallelepiped whose sides are given by $\overrightarrow{OA} = 2\hat{i} - 3\hat{j}$, $\overrightarrow{OB} = \hat{i} + \hat{j} - \hat{k}$, $\overrightarrow{OC} = 3\hat{i} - \hat{k}$ is

• A. $\frac{4}{13}$ cu unit
• B. $4$ cu unit
• C. $\frac{2}{7}$ cu unit
• D. None of these

Hint:

Volume of parallelepiped = scalar triple product.

Answer 1

The Correct Option is B

Step 1: Volume = $|\overrightarrow{OA} \cdot (\overrightarrow{OB} \times \overrightarrow{OC})|$.}
Step 2: $\overrightarrow{OB} \times \overrightarrow{OC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
1 & 1 & -1
3 & 0 & -1 \end{vmatrix} = \hat{i}(-1-0) - \hat{j}(-1+3) + \hat{k}(0-3) = -\hat{i} - 2\hat{j} - 3\hat{k}$.}
Step 3: $\overrightarrow{OA} \cdot (\overrightarrow{OB} \times \overrightarrow{OC}) = (2)(-1) + (-3)(-2) + (0)(-3) = -2+6+0=4$. So volume = $4$ cu units.}