If `|veca|=5, |vecb|=3, |vecc|=4` and `veca` is perpendicular to `vecb` and `vecc` such that angle between `vecb` and `vecc` is `(5pi)/6`, then the volume of the parallelopiped having `veca, vecb` and `vecc` as three coterminous edges is

Options

A. 30 cubit units
B. 60 cubic units
C. 20 cubic units
D. none of these

Explanation

To find the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we can use the formula for the volume \(V\) given by the scalar triple product:

\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 1: Identify the magnitudes and relationships of the vectors
We are given:
- \(|\vec{a}| = 5\)
- \(|\vec{b}| = 3\)
- \(|\vec{c}| = 4\)
- \(\vec{a}\) is perpendicular to both \(\vec{b}\) and \(\vec{c}\).

Since \(\vec{a}\) is perpendicular to \(\vec{b}\) and \(\vec{c}\), we have:
\[ \vec{a} \cdot \vec{b} = 0 \quad \text{and} \quad \vec{a} \cdot \vec{c} = 0 \]

### Step 2: Find the angle between \(\vec{b}\) and \(\vec{c}\)
The angle between \(\vec{b}\) and \(\vec{c}\) is given as:
\[ \theta = \frac{5\pi}{6} \]

### Step 3: Calculate the magnitude of \(\vec{b} \times \vec{c}\)
The magnitude of the cross product \(|\vec{b} \times \vec{c}|\) can be calculated using the formula:
\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin(\theta) \]
Substituting the known values:
\[ |\vec{b} \times \vec{c}| = 3 \times 4 \times \sin\left(\frac{5\pi}{6}\right) \]

### Step 4: Calculate \(\sin\left(\frac{5\pi}{6}\right)\)
Using the sine function:
\[ \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Step 5: Substitute back to find \(|\vec{b} \times \vec{c}|\)
Now substituting this back:
\[ |\vec{b} \times \vec{c}| = 3 \times 4 \times \frac{1}{2} = 6 \]

### Step 6: Calculate the volume of the parallelepiped
Now we can find the volume using the scalar triple product:
\[ V = |\vec{a}| \cdot |\vec{b} \times \vec{c}| = 5 \cdot 6 = 30 \]

### Final Answer
Thus, the volume of the parallelepiped is:
\[ \boxed{30} \text{ cubic units} \]