Question

If the volume of the parallelepiped whose coterminous edges are along the vectors \[ \vec{a}, \vec{b}, \vec{c} \] is 12, then the volume of the tetrahedron whose coterminous edges are \[ \vec{a} + \vec{b}, \, \vec{b} + \vec{c}, \, \vec{c} + \vec{a} \] is

• A. \( 4 \, \text{units}^3 \)
• B. \( 24 \, \text{units}^3 \)
• C. \( 6 \, \text{units}^3 \)
• D. \( 12 \, \text{units}^3 \)

Hint:

To find the volume of a tetrahedron, divide the volume of the corresponding parallelepiped by 3.

Answer 1

The Correct Option is A

Step 1: Understand the formula for volume of a parallelepiped.
The volume of a parallelepiped is given by the scalar triple product \( V = \left| \vec{a} \cdot (\vec{b} \times \vec{c}) \right| \). We are told that the volume of the parallelepiped is 12.

Step 2: Use the relation for the volume of the tetrahedron.
The volume of the tetrahedron is \( \frac{1}{6} \) of the volume of the parallelepiped. Given the volume of the parallelepiped is 12, the volume of the tetrahedron is \( \frac{12}{3} = 4 \).

Step 3: Conclusion.
Thus, the volume of the tetrahedron is 4 units³, corresponding to option (A).