A vector $$\vec{A}$$ is rotated by a small angle $$\Delta\theta$$ radians $$(\Delta\theta \ll 1)$$ to get a new vector $$\vec{B}$$. In that case $$\left|\vec{B} - \vec{A}\right|$$ is:

We begin by noting that the new vector $$\vec B$$ is obtained from the old vector $$\vec A$$ purely by rotation through a very small angle $$\Delta\theta$$ (measured in radians). Rotation does not change magnitude, so we immediately have

$$|\vec A| = |\vec B|.$$

Our task is to find the magnitude of the difference vector $$\vec B-\vec A$$. The square of this magnitude can be expressed with the dot-product formula

$$|\vec B-\vec A|^{2} = (\vec B-\vec A)\cdot(\vec B-\vec A).$$

Expanding the right-hand side using distributivity of the dot product, we obtain

$$$ |\vec B-\vec A|^{2}= \vec B\cdot\vec B + \vec A\cdot\vec A - 2\,\vec A\cdot\vec B. $$$

Recognising that $$\vec B\cdot\vec B = |\vec B|^{2}$$ and $$\vec A\cdot\vec A = |\vec A|^{2}$$, and recalling that $$\vec A\cdot\vec B = |\vec A|\,|\vec B|\cos\Delta\theta,$$ we rewrite the expression as

$$$ |\vec B-\vec A|^{2}= |\vec B|^{2}+|\vec A|^{2}-2|\vec A|\,|\vec B|\cos\Delta\theta. $$$

Because $$|\vec A|=|\vec B|,$$ let us denote this common magnitude simply by $$|\vec A|$$. Substituting, we have

$$$ |\vec B-\vec A|^{2}= |\vec A|^{2}+|\vec A|^{2}-2|\vec A|^{2}\cos\Delta\theta = 2|\vec A|^{2}\bigl(1-\cos\Delta\theta\bigr). $$$

Now we employ the small-angle approximation for cosine. For very small $$\Delta\theta$$ (in radians), the Taylor series gives

$$$ \cos\Delta\theta \approx 1-\frac{(\Delta\theta)^{2}}{2}. $$$

Substituting this approximation into the previous expression, we find

$$$ |\vec B-\vec A|^{2} \approx 2|\vec A|^{2}\left[1-\left(1-\frac{(\Delta\theta)^{2}}{2}\right)\right] = 2|\vec A|^{2}\left[\frac{(\Delta\theta)^{2}}{2}\right] = |\vec A|^{2}(\Delta\theta)^{2}. $$$

Taking the square root of both sides (and keeping only the positive root because a magnitude is always positive) gives

$$$ |\vec B-\vec A| \approx |\vec A|\,\Delta\theta. $$$

This expression is exactly Option C in the list provided. None of the other options match the derived result.

Hence, the correct answer is Option C.