Let `bara` be a unit vector perpendicular to unit vectors `barb and barc` inclined at an angle a. then `barb times barc` is

Options

A. `(cosa)bara`
B. `(coseca)bara`
C. `(sina)bara`
D. None of these

Answer

To solve the problem, we need to find the value of the cross product of two unit vectors \(\mathbf{b}\) and \(\mathbf{c}\) that are inclined at an angle \(a\). Additionally, we know that there exists a unit vector \(\mathbf{a}\) that is perpendicular to both \(\mathbf{b}\) and \(\mathbf{c}\).

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- Let \(\mathbf{b}\) and \(\mathbf{c}\) be unit vectors.
- Since they are unit vectors, we have:
\[ |\mathbf{b}| = 1 \quad \text{and} \quad |\mathbf{c}| = 1 \]

2. **Using the Cross Product Formula**:
- The formula for the cross product of two vectors \(\mathbf{b}\) and \(\mathbf{c}\) is given by:
\[ \mathbf{b} \times \mathbf{c} = |\mathbf{b}| |\mathbf{c}| \sin(\theta) \, \mathbf{n} \]
where \(\theta\) is the angle between the vectors and \(\mathbf{n}\) is the unit vector perpendicular to the plane formed by \(\mathbf{b}\) and \(\mathbf{c}\).

3. **Substituting Values**:
- Since both \(\mathbf{b}\) and \(\mathbf{c}\) are unit vectors, we have:
\[ |\mathbf{b}| = 1 \quad \text{and} \quad |\mathbf{c}| = 1 \]
- The angle \(\theta\) between \(\mathbf{b}\) and \(\mathbf{c}\) is given as \(a\).
- Therefore, we can substitute these values into the formula:
\[ \mathbf{b} \times \mathbf{c} = 1 \cdot 1 \cdot \sin(a) \, \mathbf{n} = \sin(a) \, \mathbf{n} \]

4. **Identifying the Perpendicular Vector**:
- The vector \(\mathbf{n}\) is the unit vector that is perpendicular to both \(\mathbf{b}\) and \(\mathbf{c}\). We are given that this vector is denoted as \(\mathbf{a}\).
- Thus, we can write:
\[ \mathbf{b} \times \mathbf{c} = \sin(a) \, \mathbf{a} \]

5. **Final Result**:
- Therefore, the value of \(\mathbf{b} \times \mathbf{c}\) is:
\[ \mathbf{b} \times \mathbf{c} = \sin(a) \, \mathbf{a} \]

### Conclusion:
The final answer is:
\[ \mathbf{b} \times \mathbf{c} = \sin(a) \, \mathbf{a} \]