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: Recall the formula for the volume of the parallelepiped.
The volume \( V \) of a parallelepiped formed by three vectors \( \vec{a}, \vec{b}, \vec{c} \) is given by the scalar triple product: \[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})|. \]
Step 2: Compute the cross product \( \vec{b} \times \vec{c} \).
Let \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} - 4\hat{j} - \hat{k} \). First, compute the cross product \( \vec{b} \times \vec{c} \). Then, take the dot product of \( \vec{a} \) with this result.
Step 3: Find the result of the scalar triple product.
After calculating, we get the volume as 8 cubic units.

Step 4: Conclusion.
Thus, the volume of the parallelepiped is 8, which corresponds to option (A).