Question

If the volume of a tetrahedron whose conterminous edges are $\vec{\mathrm{a}}+\vec{\mathrm{b}}$, $\vec{\mathrm{b}}+\vec{\mathrm{c}}$, $\vec{\mathrm{c}}+\vec{\mathrm{a}}$ is 24 cubic units, then the volume of parallelopiped whose coterminous edges are $\vec{\mathrm{a}}$, $\vec{\mathrm{b}}$, $\vec{\mathrm{c}}$ is

• A. 48 cubic units
• B. 144 cubic units
• C. 72 cubic units
• D. 10 cubic units

Hint:

Remember the standard scaling multipliers: $\text{Volume of Tetrahedron}(\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{c}+\vec{a}) = \frac{2}{6}[\vec{a}\ \vec{b}\ \vec{c}] = \frac{1}{3}\text{Volume of Parallelopiped}(\vec{a}, \vec{b}, \vec{c})$. So, simply multiply the given tetrahedron volume by 3: $24 \times 3 = 72$!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem relates the volume of a tetrahedron formed by the vectors $\vec{a}+\vec{b}$, $\vec{b}+\vec{c}$, and $\vec{c}+\vec{a}$ to the volume of a parallelopiped formed by the individual vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.

Step 2: Key Formula or Approach:
The volume of a tetrahedron with conterminous edges $\vec{u}$, $\vec{v}$, $\vec{w}$ is given by $\frac{1}{6}|[\vec{u}\ \vec{v}\ \vec{w}]|$, where $[\vec{u}\ \vec{v}\ \vec{w}]$ represents the scalar triple product. The volume of a parallelopiped with conterminous edges $\vec{a}$, $\vec{b}$, $\vec{c}$ is given directly by the absolute value of their scalar triple product, $|[\vec{a}\ \vec{b}\ \vec{c}]|$. A well-known vector identity states that: $$ [\vec{a}+\vec{b} \quad \vec{b}+\vec{c} \quad \vec{c}+\vec{a}] = 2[\vec{a}\ \vec{b}\ \vec{c}] $$

Step 3: Detailed Explanation:
Given that the volume of the tetrahedron is 24 cubic units: $$ \frac{1}{6} |[\vec{a}+\vec{b} \quad \vec{b}+\vec{c} \quad \vec{c}+\vec{a}]| = 24 $$ Multiplying both sides by 6: $$ |[\vec{a}+\vec{b} \quad \vec{b}+\vec{c} \quad \vec{c}+\vec{a}]| = 144 $$ Now, substituting the scalar triple product identity into the equation: $$ 2|[\vec{a}\ \vec{b}\ \vec{c}]| = 144 $$ Dividing by 2 to isolate the parallelopiped volume expression: $$ |[\vec{a}\ \vec{b}\ \vec{c}]| = \frac{144}{2} = 72\ \text{cubic units} $$

Step 4: Final Answer:
The volume of the parallelopiped is 72 cubic units, which corresponds to option (C).