Let ` veca ` be a unit vector perpendicular to unit vectors `vecb and vecc` and if the angle between `vecb and vecc " is " alpha , " then " vecb xx vecc` is

Options

A. ` +- ( cos alpha ) veca`
B. ` +- ( cosec alpha) veca`
C. `+- ( sin alpha) veca`
D. none of these

Explanation

To solve the problem, we need to find the vector \( \vec{b} \times \vec{c} \) given that \( \vec{a} \) is a unit vector perpendicular to both \( \vec{b} \) and \( \vec{c} \), and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \alpha \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- \( \vec{a} \) is a unit vector, meaning \( |\vec{a}| = 1 \).
- \( \vec{b} \) and \( \vec{c} \) are also unit vectors, so \( |\vec{b}| = 1 \) and \( |\vec{c}| = 1 \).
- The angle between \( \vec{b} \) and \( \vec{c} \) is \( \alpha \).

2. **Using the Property of Cross Product:**
- Since \( \vec{a} \) is perpendicular to both \( \vec{b} \) and \( \vec{c} \), we can express \( \vec{a} \) in terms of the cross product of \( \vec{b} \) and \( \vec{c} \):
\[ \vec{a} = k (\vec{b} \times \vec{c}) \]
where \( k \) is a scalar.

3. **Finding the Magnitude of the Cross Product:**
- The magnitude of the cross product \( |\vec{b} \times \vec{c}| \) can be calculated using the formula:
\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin(\alpha) \]
- Since both \( \vec{b} \) and \( \vec{c} \) are unit vectors, we have:
\[ |\vec{b} \times \vec{c}| = 1 \cdot 1 \cdot \sin(\alpha) = \sin(\alpha) \]

4. **Expressing \( \vec{a} \):**
- Since \( \vec{a} \) is a unit vector, we can write:
\[ |\vec{a}| = |k| |\vec{b} \times \vec{c}| \]
- Given that \( |\vec{a}| = 1 \) and \( |\vec{b} \times \vec{c}| = \sin(\alpha) \), we have:
\[ 1 = |k| \sin(\alpha) \]
- Therefore, \( |k| = \frac{1}{\sin(\alpha)} \).

5. **Final Expression for \( \vec{b} \times \vec{c} \):**
- Thus, we can express \( \vec{b} \times \vec{c} \) as:
\[ \vec{b} \times \vec{c} = \pm \vec{a} \sin(\alpha) \]

### Conclusion:
The final result for \( \vec{b} \times \vec{c} \) is:
\[ \vec{b} \times \vec{c} = \pm \vec{a} \sin(\alpha) \]