Question

The volume of the parallelepiped whose co-terminous edges are $\hat{i} + \hat{j}, \hat{i} + \hat{k}, \hat{i} + \hat{j}$ is:

• A. $6$ cu. units
• B. $2$ cu. units
• C. $4$ cu. units
• D. $3$ cu. units

Answer 1

The Correct Option is B

The volume of a parallelepiped is given by the scalar triple product: \[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \] Where: - $\vec{a} = \hat{i} + \hat{j}$ - $\vec{b} = \hat{i} + \hat{k}$ - $\vec{c} = \hat{i} + \hat{j}$ First, compute the cross product $\vec{b} \times \vec{c}$: \[ \vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{vmatrix} = \hat{i}(-1) - \hat{j}(1) + \hat{k}(1) = -\hat{i} - \hat{j} + \hat{k} \] Next, compute the dot product $\vec{a} \cdot (\vec{b} \times \vec{c})$: \[ \vec{a} \cdot (\vec{b} \times \vec{c}) = (1)(-1) + (1)(-1) + (0)(1) = -2 \] Thus, the volume is: \[ V = | -2 | = 2 \, \text{cu. units} \] Hence, the correct answer is $2$ cu. units.