Let `veca, vecb, vecc` be three unit vectors such that `veca. vecb=veca.vecc=0`, If the angle between `vecb` and `vecc` is `(pi)/3` then the volume of the parallelopiped whose three coterminous edges are `veca, vecb, vecc` is

Options

A. `(sqrt(3))/2` cubic units
B. `1/2` cubit unit
C. 1 cubic unit
D. none of these

Explanation

To find the volume of the parallelepiped formed by the three unit vectors \(\vec{a}, \vec{b}, \vec{c}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have three unit vectors \(\vec{a}, \vec{b}, \vec{c}\) such that \(\vec{a} \cdot \vec{b} = 0\) and \(\vec{a} \cdot \vec{c} = 0\). This means that \(\vec{a}\) is perpendicular to both \(\vec{b}\) and \(\vec{c}\). The angle between \(\vec{b}\) and \(\vec{c}\) is given as \(\frac{\pi}{3}\).

2. **Volume of the Parallelepiped**: The volume \(V\) of the parallelepiped formed by the vectors \(\vec{a}, \vec{b}, \vec{c}\) can be calculated using the formula:
\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

3. **Magnitude of the Vectors**: Since \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors, we have:
\[ |\vec{a}| = |\vec{b}| = |\vec{c}| = 1 \]

4. **Finding \(\vec{b} \times \vec{c}\)**: The magnitude of the cross product \(\vec{b} \times \vec{c}\) can be calculated using the formula:
\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin(\theta) \]
where \(\theta\) is the angle between \(\vec{b}\) and \(\vec{c}\). Given that \(\theta = \frac{\pi}{3}\):
\[ |\vec{b} \times \vec{c}| = 1 \cdot 1 \cdot \sin\left(\frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

5. **Direction of \(\vec{b} \times \vec{c}\)**: Since \(\vec{a}\) is perpendicular to both \(\vec{b}\) and \(\vec{c}\), \(\vec{a}\) is parallel to \(\vec{b} \times \vec{c}\). Therefore, we can write:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = |\vec{a}| |\vec{b} \times \vec{c}| = 1 \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

6. **Final Volume Calculation**: Thus, the volume of the parallelepiped is:
\[ V = \left|\vec{a} \cdot (\vec{b} \times \vec{c})\right| = \frac{\sqrt{3}}{2} \]

7. **Conclusion**: The volume of the parallelepiped formed by the vectors \(\vec{a}, \vec{b}, \vec{c}\) is:
\[ \frac{\sqrt{3}}{2} \text{ cubic units} \]