If A is a unit vector in a given direction then the value of `hatA.(dhatA)/(dt)` is

Options

A. 0
B. 1
C. `(1)/(2)`
D. 2

Explanation

To solve the problem, we need to determine the value of the expression \(\hat{A} \cdot \frac{d\hat{A}}{dt}\) where \(\hat{A}\) is a unit vector.

### Step-by-Step Solution:

1. **Understanding Unit Vectors**:
A unit vector \(\hat{A}\) is defined as a vector with a magnitude of 1. This means that \(|\hat{A}| = 1\).

2. **Differentiating the Unit Vector**:
Since \(\hat{A}\) is a unit vector, its magnitude is constant. Therefore, when we differentiate \(\hat{A}\) with respect to time \(t\), we can use the property of unit vectors.

3. **Using the Derivative of Magnitude**:
The magnitude of \(\hat{A}\) can be expressed mathematically as:
\[ |\hat{A}|^2 = \hat{A} \cdot \hat{A} = 1 \]
Differentiating both sides with respect to \(t\):
\[ \frac{d}{dt}(\hat{A} \cdot \hat{A}) = 0 \]
Using the product rule, we get:
\[ 2 \hat{A} \cdot \frac{d\hat{A}}{dt} = 0 \]

4. **Solving for the Dot Product**:
From the equation \(2 \hat{A} \cdot \frac{d\hat{A}}{dt} = 0\), we can simplify this to:
\[ \hat{A} \cdot \frac{d\hat{A}}{dt} = 0 \]
This indicates that the dot product of the unit vector \(\hat{A}\) and its derivative with respect to time is zero.

5. **Conclusion**:
Therefore, the value of \(\hat{A} \cdot \frac{d\hat{A}}{dt}\) is:
\[ \hat{A} \cdot \frac{d\hat{A}}{dt} = 0 \]

### Final Answer:
The value of \(\hat{A} \cdot \frac{d\hat{A}}{dt}\) is **0**.