Question

If \(\vec u = \hat i - 2\hat j + \hat k\), \(\vec v = 3\hat i + \hat k\) and \(\vec w = \hat j - \hat k\), then the volume of the parallelepiped with \(\vec u \times \vec v\), \(\vec u + \vec w\) and \(\vec v + \vec w\) as coterminous edges is

• A. 12 cubic units
• B. 10 cubic units
• C. 24 cubic units
• D. 20 cubic units

Hint:

The volume of a parallelepiped is always given by the absolute value of a scalar triple product.

Answer 1

The Correct Option is C

Step 1: Use the formula for volume of a parallelepiped.
The volume is given by \[ V = \left| (\vec u \times \vec v) \cdot (\vec u + \vec w \times \vec v + \vec w) \right| \] Equivalently, the volume is the absolute value of the scalar triple product.

Step 2: Find \(\vec u \times \vec v\).
\[ \vec u = \begin{vmatrix} \hat i & \hat j & \hat k \\ 1 & -2 & 1 \end{vmatrix}, \quad \vec v = \begin{vmatrix} 3 & 0 & 1 \end{vmatrix} \] \[ \vec u \times \vec v = \begin{vmatrix} \hat i & \hat j & \hat k \\ 1 & -2 & 1 \\ 3 & 0 & 1 \end{vmatrix} = (-2)\hat i + 2\hat j + 6\hat k \]
Step 3: Find \(\vec u + \vec w\) and \(\vec v + \vec w\).
\[ \vec u + \vec w = (1, -1, 0), \quad \vec v + \vec w = (3, 1, 0) \]
Step 4: Compute the scalar triple product.
\[ V = \left| \begin{vmatrix} -2 & 2 & 6 \\ 1 & -1 & 0\\ 3 & 1 & 0 \end{vmatrix} \right| = |24| \]