The volume of a parallelopiped whose sides are given by `vecOA =2i-3j,vecOB =i+j-k,vecOC =3i-k` is

Options

A. `4//13`
B. 4
C. `2//7`
D. none of these

Answer

To find the volume of the parallelepiped formed by the vectors \(\vec{OA}\), \(\vec{OB}\), and \(\vec{OC}\), we can use the scalar triple product, which is given by the determinant of a matrix formed by the components of these vectors.

Given:
- \(\vec{OA} = 2\hat{i} - 3\hat{j}\)
- \(\vec{OB} = \hat{i} + \hat{j} - \hat{k}\)
- \(\vec{OC} = 3\hat{i} - \hat{k}\)

### Step 1: Write the vectors in matrix form
We can represent the vectors as follows:
\[ \begin{bmatrix} \vec{OA} \\ \vec{OB} \\ \vec{OC} \end{bmatrix} = \begin{bmatrix} 2 & -3 & 0 \\ 1 & 1 & -1 \\ 3 & 0 & -1 \end{bmatrix} \]

### Step 2: Set up the determinant
The volume \(V\) of the parallelepiped is given by the absolute value of the determinant of the matrix formed by the vectors:
\[ V = \left| \begin{vmatrix} 2 & -3 & 0 \\ 1 & 1 & -1 \\ 3 & 0 & -1 \end{vmatrix} \right| \]

### Step 3: Calculate the determinant
To calculate the determinant, we can use the formula for a \(3 \times 3\) matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]
In our case:
- \(a = 2\), \(b = -3\), \(c = 0\)
- \(d = 1\), \(e = 1\), \(f = -1\)
- \(g = 3\), \(h = 0\), \(i = -1\)

Calculating the determinant:
\[ \text{det}(A) = 2(1 \cdot (-1) - (-1) \cdot 0) - (-3)(1 \cdot (-1) - (-1) \cdot 3) + 0(1 \cdot 0 - 1 \cdot 3) \]
\[ = 2(-1 - 0) + 3(-1 + 3) \]
\[ = 2(-1) + 3(2) \]
\[ = -2 + 6 \]
\[ = 4 \]

### Step 4: Find the absolute value
Since volume cannot be negative, we take the absolute value:
\[ V = |4| = 4 \]

### Conclusion
The volume of the parallelepiped is \(4\).