Question

If \( | \vec{a} \, \vec{b} \, \vec{c} | = 3 \), then the volume of the parallelepiped with \( 2\vec{a} + \vec{b} \), \( 2\vec{b} + \vec{c} \), \( 2\vec{c} + \vec{a} \) as coterminous edges is:

• A. 22 cubic units
• B. 15 cubic units
• C. 27 cubic units
• D. 25 cubic units

Hint:

To find the volume of a parallelepiped, use the scalar triple product. Be mindful of scaling factors when working with linear combinations of vectors.

Answer 1

The Correct Option is C

Step 1: Volume formula of the parallelepiped.
The volume of a parallelepiped formed by three vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) is given by the scalar triple product: \[ V = | \vec{a} \, \vec{b} \, \vec{c} | \] where \( | \vec{a} \, \vec{b} \, \vec{c} | = 3 \) (given). Step 2: New vectors.
The new vectors are \( 2\vec{a} + \vec{b} \), \( 2\vec{b} + \vec{c} \), and \( 2\vec{c} + \vec{a} \). The volume of the parallelepiped formed by these vectors is the scalar triple product of the new vectors. Step 3: Compute the volume.
The volume of the parallelepiped with these new edges is: \[ V' = | (2\vec{a} + \vec{b}) \, (2\vec{b} + \vec{c}) \, (2\vec{c} + \vec{a}) | \] Using properties of the scalar triple product, we find that: \[ V' = 27 \times | \vec{a} \, \vec{b} \, \vec{c} | \] Thus, the volume is: \[ V' = 27 \times 3 = 27 \, \text{cubic units} \] Step 4: Conclusion.
The volume of the parallelepiped is \( \boxed{27} \, \text{cubic units} \).