One vertex of a parallelopiped is at the point `(1, -1, -2)` of rectangular cartesian coordinates. If three adjacent vertices are at (0, 1, 3), (3, 0 -1) and (1, 4, 1) the volume of the parallelopiped is

Options

A. 400 unit
B. 80 unit
C. 40 unit
D. 120 unit

Answer

To find the volume of the parallelepiped formed by the given vertices, we can follow these steps:

### Step 1: Identify the vertices
Let:
- Vertex O = (1, -1, -2)
- Vertex A = (0, 1, 3)
- Vertex B = (3, 0, -1)
- Vertex C = (1, 4, 1)

### Step 2: Calculate the vectors for the adjacent sides
We need to find the vectors OA, OB, and OC.

1. **Vector OA**:
\[ \text{OA} = A - O = (0, 1, 3) - (1, -1, -2) = (0 - 1, 1 - (-1), 3 - (-2)) = (-1, 2, 5) \]
In vector form:
\[ \text{OA} = -\hat{i} + 2\hat{j} + 5\hat{k} \]

2. **Vector OB**:
\[ \text{OB} = B - O = (3, 0, -1) - (1, -1, -2) = (3 - 1, 0 - (-1), -1 - (-2)) = (2, 1, 1) \]
In vector form:
\[ \text{OB} = 2\hat{i} + \hat{j} + \hat{k} \]

3. **Vector OC**:
\[ \text{OC} = C - O = (1, 4, 1) - (1, -1, -2) = (1 - 1, 4 - (-1), 1 - (-2)) = (0, 5, 3) \]
In vector form:
\[ \text{OC} = 0\hat{i} + 5\hat{j} + 3\hat{k} \]

### Step 3: Set up the determinant for the volume
The volume \( V \) of the parallelepiped is given by the scalar triple product, which can be calculated using the determinant of the matrix formed by the vectors OA, OB, and OC:
\[ V = |\text{OA} \cdot (\text{OB} \times \text{OC})| = \begin{vmatrix} -1 & 2 & 5 \\ 2 & 1 & 1 \\ 0 & 5 & 3 \end{vmatrix} \]

### Step 4: Calculate the determinant
To compute the determinant, we can use the formula:
\[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Where:
- \( a = -1, b = 2, c = 5 \)
- \( d = 2, e = 1, f = 1 \)
- \( g = 0, h = 5, i = 3 \)

Calculating the determinant:
\[ \text{det} = -1(1 \cdot 3 - 1 \cdot 5) - 2(2 \cdot 3 - 1 \cdot 0) + 5(2 \cdot 5 - 1 \cdot 0) \]
\[ = -1(3 - 5) - 2(6 - 0) + 5(10 - 0) \]
\[ = -1(-2) - 2(6) + 5(10) \]
\[ = 2 - 12 + 50 \]
\[ = 40 \]

### Step 5: Conclusion
The volume of the parallelepiped is \( 40 \) cubic units.

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