Question

The volume of a parallelopiped with coterminous edges \(\overset{̄}{a},\overset{̄}{b},\overset{̄}{c}\) is 3 cubic units. The volume (in cubic units) of a tetrahedron with coterminous edges \((\overset{̄}{a}\times \overset{̄}{b}),(\overset{̄}{a}\times 2\overset{̄}{c}),(\overset{̄}{b}\times 2\overset{̄}{c})\) is...

• A. \(6\)
• B. \(12\)
• C. \(24\)
• D. \(36\)

Hint:

Use [a x b, b x c, c x a] = [a b c] squared, then adjust for the factors of 2.

Answer 1

The Correct Option is A

Step 1: Scale factors
Volume of a parallelepiped scales with each edge. The factors 2 and 2 multiply the volume by 4.

Step 2: Base value
The triple product of the three cross products is $[\vec a\ \vec b\ \vec c]^2 = 9$ in magnitude.

Step 3: Tetrahedron
Parallelepiped volume $= 4\times9 = 36$, tetrahedron $= \frac{36}{6} = 6$. Option (A).

Final Answer:
6. \[ \boxed{\text{(A)}\ 6} \]