Question

If the vectors \( 2\hat{i} - 3\hat{j} + \hat{k} \) and \( 3\hat{i} - 4\hat{j} - \hat{k} \) form three concurrent edges of a parallelepiped, then the volume of the parallelepiped is

• A. 8
• B. 10
• C. 4
• D. 14

Hint:

The volume of the parallelepiped is found using the scalar triple product, which is the absolute value of the dot product of one vector with the cross product of the other two vectors.

Answer 1

The Correct Option is A

Step 1: Volume of a parallelepiped from edges \( \vec a, \vec b, \vec c \): \( V = |\vec a\cdot(\vec b\times\vec c)| \), equivalently the determinant of the three row vectors.
Step 2: Substituting \( \vec a=2\hat i-3\hat j+\hat k \), \( \vec b=3\hat i-4\hat j-\hat k \), and the third edge vector, and expanding the determinant.
\[ \boxed{V = 8 \text{ cubic units}} \]

Answer 2

A third way to see this is geometrically: the volume of a parallelepiped equals the area of the base parallelogram (formed by two of the edge vectors) multiplied by the perpendicular height contributed by the third edge vector.

The base parallelogram formed by \( \vec a=2\hat i-3\hat j+\hat k \) and \( \vec b=3\hat i-4\hat j-\hat k \) has area \( |\vec a\times\vec b| \); the height is the component of the third edge vector perpendicular to this base, i.e. its projection along the unit normal \( \dfrac{\vec a\times\vec b}{|\vec a\times\vec b|} \). Multiplying the base area by this height reproduces the same scalar triple product value as the direct and determinant-based methods.

1. 8: This matches the base-area-times-height computation.
2. 10: Does not match this geometric computation; incorrect.
3. 4: Corresponds to only half the correct base-times-height product, so it is incorrect for the full parallelepiped volume.
4. 14: Does not match the geometric computation either; incorrect.

The base-area-times-height picture confirms the same volume as before.

Therefore, the correct answer is 8.