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: Rewrite the edges
The edges are \(\vec a\times\vec b\), \(2(\vec a\times\vec c)\) and \(2(\vec b\times\vec c)\).

Step 2: Triple product
The scalar triple product is \(4\,[\vec a\times\vec b,\ \vec a\times\vec c,\ \vec b\times\vec c]\).

Step 3: Use the identity
\([\vec a\times\vec b,\ \vec b\times\vec c,\ \vec c\times\vec a] = [\vec a\ \vec b\ \vec c]^2 = 9\). Our triple uses \(\vec a\times\vec c = -(\vec c\times\vec a)\), and its factors are in the order \(\vec a\times\vec b,\ \vec c\times\vec a,\ \vec b\times\vec c\) after that, which is one swap from the cyclic order. The two sign changes cancel, so the triple product is \(9\). With the factors of 2, the triple product is \(4\times9 = 36\).

Step 4: Tetrahedron
The volume of the tetrahedron is \(\frac16\times36 = 6\). Option (A).

Final Answer:
The tetrahedron volume is 6 cubic units. \[ \boxed{\text{(A)}\ 6} \]