Let the unit vectors `a` and `b` be perpendicular and the unit vector `c` be inclined at an angle `theta` to both `a` and `b`. If `c=alpha a+beta b+gamma(axxb)`, then

Options

A. `alpha=beta=costheta`
B. `alpha=costheta,beta=sintheta`
C. `gamma^(2)=cos2theta`
D. `gamma^(2)=-cos2theta`

Explanation

To solve the question, we need to find the values of α, β, and γ in the expression \( \mathbf{c} = \alpha \mathbf{a} + \beta \mathbf{b} + \gamma (\mathbf{a} \times \mathbf{b}) \), given that the unit vectors \( \mathbf{a} \) and \( \mathbf{b} \) are perpendicular and the unit vector \( \mathbf{c} \) is inclined at an angle \( \theta \) to both \( \mathbf{a} \) and \( \mathbf{b} \).

### Step-by-Step Solution:

1. **Understanding the Properties of Unit Vectors**:
- Since \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, we have:
\[
|\mathbf{a}| = 1 \quad \text{and} \quad |\mathbf{b}| = 1
...