Let `hat(u), hat(v) and hat(w)` are three unit vectors, the angle between `hat(u) and hat(v)` is twice that of the angle between `hat(u) and hat(w) and hat(v) and hat(w)`, then `[hat(u) hat(v) hat(w)]` is equal to

Answer

To solve the problem, we need to find the scalar triple product \([ \hat{u}, \hat{v}, \hat{w} ]\) given that the angle between \(\hat{u}\) and \(\hat{v}\) is twice that of the angle between \(\hat{u}\) and \(\hat{w}\) and the angle between \(\hat{v}\) and \(\hat{w}\).

### Step-by-step Solution:

1. **Understanding the Angles**:
Let the angle between \(\hat{u}\) and \(\hat{w}\) be \(\theta\). Then, the angle between \(\hat{u}\) and \(\hat{v}\) will be \(2\theta\), and the angle between \(\hat{v}\) and \(\hat{w}\) will also be \(\theta\).

2. **Using the Dot Product**:
Since \(\hat{u}\), \(\hat{v}\), and \(\hat{w}\) are unit vectors, we can express the dot products as follows:
\[ \hat{u} \cdot \hat{u} = 1, \]
\[ \hat{v} \cdot \hat{v} = 1, \]
\[ \hat{w} \cdot \hat{w} = 1, \]
\[ \hat{u} \cdot \hat{v} = \cos(2\theta), \]
\[ \hat{u} \cdot \hat{w} = \cos(\theta), \]
\[ \hat{v} \cdot \hat{w} = \cos(\theta). \]

3. **Setting Up the Determinant**:
The scalar triple product can be represented as the determinant of a matrix formed by the dot products:
\[ [\hat{u}, \hat{v}, \hat{w}]^2 = \begin{vmatrix} \hat{u} \cdot \hat{u} & \hat{u} \cdot \hat{v} & \hat{u} \cdot \hat{w} \\ \hat{v} \cdot \hat{u} & \hat{v} \cdot \hat{v} & \hat{v} \cdot \hat{w} \\ \hat{w} \cdot \hat{u} & \hat{w} \cdot \hat{v} & \hat{w} \cdot \hat{w} \end{vmatrix} \]

Substituting the values we have:
\[ = \begin{vmatrix} 1 & \cos(2\theta) & \cos(\theta) \\ \cos(2\theta) & 1 & \cos(\theta) \\ \cos(\theta) & \cos(\theta) & 1 \end{vmatrix} \]

4. **Calculating the Determinant**:
We can expand this determinant. The determinant can be calculated using the formula for a 3x3 matrix:
\[ = 1 \cdot (1 \cdot 1 - \cos^2(\theta)) - \cos(2\theta) \cdot (\cos(2\theta) \cdot 1 - \cos(\theta) \cdot \cos(\theta)) + \cos(\theta) \cdot (\cos(2\theta) \cdot \cos(\theta) - \cos(\theta) \cdot 1) \]

After simplifying, we find that the determinant equals zero.

5. **Conclusion**:
Since the square of the scalar triple product is zero, we conclude that:
\[ [\hat{u}, \hat{v}, \hat{w}] = 0. \]

### Final Answer:
\[ [\hat{u}, \hat{v}, \hat{w}] = 0. \]