If $$\hat{a}$$, $$\hat{b}$$ and $$\hat{c}$$ are unit vectors satisfying $$\hat{a} - \sqrt{3}\hat{b} + \hat{c} = \vec{0}$$, then the angle between the vectors $$\hat{a}$$ and $$\hat{c}$$ is :

We are given that $$\hat{a}$$, $$\hat{b}$$, and $$\hat{c}$$ are unit vectors, so their magnitudes are all 1. The equation provided is:

$$\hat{a} - \sqrt{3}\hat{b} + \hat{c} = \vec{0}$$

Rearranging this equation, we get:

$$\hat{a} + \hat{c} = \sqrt{3}\hat{b}$$

Since $$\hat{b}$$ is a unit vector, $$|\hat{b}| = 1$$. Therefore, the magnitude of the right side is:

$$|\sqrt{3}\hat{b}| = \sqrt{3} |\hat{b}| = \sqrt{3} \times 1 = \sqrt{3}$$

Thus, the magnitude of the left side must also be $$\sqrt{3}$$:

$$|\hat{a} + \hat{c}| = \sqrt{3}$$

Now, we compute the square of the magnitude of $$\hat{a} + \hat{c}$$:

$$|\hat{a} + \hat{c}|^2 = (\hat{a} + \hat{c}) \cdot (\hat{a} + \hat{c})$$

Expanding the dot product:

$$|\hat{a} + \hat{c}|^2 = \hat{a} \cdot \hat{a} + 2(\hat{a} \cdot \hat{c}) + \hat{c} \cdot \hat{c}$$

Since $$\hat{a}$$ and $$\hat{c}$$ are unit vectors, $$\hat{a} \cdot \hat{a} = |\hat{a}|^2 = 1^2 = 1$$ and $$\hat{c} \cdot \hat{c} = |\hat{c}|^2 = 1^2 = 1$$. Substituting these values:

$$|\hat{a} + \hat{c}|^2 = 1 + 2(\hat{a} \cdot \hat{c}) + 1 = 2 + 2(\hat{a} \cdot \hat{c})$$

But we know that $$|\hat{a} + \hat{c}| = \sqrt{3}$$, so:

$$|\hat{a} + \hat{c}|^2 = (\sqrt{3})^2 = 3$$

Setting the expressions equal:

$$2 + 2(\hat{a} \cdot \hat{c}) = 3$$

Subtracting 2 from both sides:

$$2(\hat{a} \cdot \hat{c}) = 1$$

Dividing both sides by 2:

$$\hat{a} \cdot \hat{c} = \frac{1}{2}$$

The dot product of two vectors is given by the product of their magnitudes and the cosine of the angle between them. Since both are unit vectors:

$$\hat{a} \cdot \hat{c} = |\hat{a}| |\hat{c}| \cos \theta = 1 \times 1 \times \cos \theta = \cos \theta$$

So:

$$\cos \theta = \frac{1}{2}$$

The angle $$\theta$$ between $$\hat{a}$$ and $$\hat{c}$$ that satisfies this in the range $$[0, \pi]$$ is $$\theta = \frac{\pi}{3}$$.

Hence, the angle between the vectors $$\hat{a}$$ and $$\hat{c}$$ is $$\frac{\pi}{3}$$.

Comparing with the options:

A. $$\frac{\pi}{4}$$

B. $$\frac{\pi}{3}$$

C. $$\frac{\pi}{6}$$

D. $$\frac{\pi}{2}$$

So, the correct answer is Option B.