To solve the problem, we need to determine the unit vector in the direction of the friction force acting on the car as it moves along the circular path defined by the equation \(x^2 + y^2 = a^2\). The car is at the point \((a \cos \theta, a \sin \theta)\).
### Step-by-Step Solution:
1. **Understanding the Motion**: The car is moving in a circular path with a constant speed in the anti-clockwise direction. The position of the car at any instant is given by the coordinates \((a \cos \theta, a \sin \theta)\).
2. **Centripetal Force**: Since the car is moving in a circle, it requires a centripetal force to keep it in circular motion. In this case, the friction force acts as the centripetal force.
3. **Direction of the Friction Force**: The friction force must act towards the center of the circle to provide the necessary centripetal force. The center of the circle is at the origin \((0, 0)\).
4. **Finding the Vector from the Point to the Center**: The position vector of the car at the point \((a \cos \theta, a \sin \theta)\) can be represented as:
\[
\mathbf{r} = a \cos \theta \, \hat{i} + a \sin \theta \, \hat{j}
\]
The vector pointing from the car to the center (origin) is:
\[
\mathbf{F}_{\text{friction}} = 0 - (a \cos \theta \, \hat{i} + a \sin \theta \, \hat{j}) = -a \cos \theta \, \hat{i} - a \sin \theta \, \hat{j}
\]
5. **Finding the Unit Vector**: To find the unit vector in the direction of the friction force, we need to divide the friction force vector by its magnitude. The magnitude of the friction force vector is:
\[
|\mathbf{F}_{\text{friction}}| = \sqrt{(-a \cos \theta)^2 + (-a \sin \theta)^2} = \sqrt{a^2 (\cos^2 \theta + \sin^2 \theta)} = a
\]
Therefore, the unit vector in the direction of the friction force is:
\[
\hat{F}_{\text{friction}} = \frac{\mathbf{F}_{\text{friction}}}{|\mathbf{F}_{\text{friction}}|} = \frac{-a \cos \theta \, \hat{i} - a \sin \theta \, \hat{j}}{a} = -\cos \theta \, \hat{i} - \sin \theta \, \hat{j}
\]
6. **Final Result**: Thus, the unit vector in the direction of the friction force acting on the car is:
\[
\hat{F}_{\text{friction}} = -\cos \theta \, \hat{i} - \sin \theta \, \hat{j}
\]
