Question

If $\vec a,\vec b,\vec c$ are nonzero vectors along the coterminous edges of a parallelepiped with volume $7$ cubic units, then the volume of the parallelepiped with $\vec a+\vec b,\ \vec b+\vec c,\ \vec c+\vec a$ as the coterminous edges is

• A. $49$ cubic units
• B. $2$ cubic units
• C. $14$ cubic units
• D. $7$ cubic units

Hint:

Vector identities help reduce complicated volume expressions quickly.

Answer 1

The Correct Option is C

Step 1: Using scalar triple product.
The volume of a parallelepiped formed by vectors $\vec p,\vec q,\vec r$ is \[ |\vec p\cdot(\vec q\times\vec r)| \]
Step 2: Writing the new volume expression.
\[ V=|(\vec a+\vec b)\cdot[(\vec b+\vec c)\times(\vec c+\vec a)]| \]
Step 3: Expanding the cross product.
\[ (\vec b+\vec c)\times(\vec c+\vec a) =\vec b\times\vec c+\vec b\times\vec a+\vec c\times\vec a \]
Step 4: Using properties of scalar triple product.
\[ V=2|\vec a\cdot(\vec b\times\vec c)| \] Given original volume $=7$.
Step 5: Conclusion.
\[ V=2\times7=14\ \text{cubic units} \]