If $$\vec{A}$$ and $$\vec{B}$$ are two vectors satisfying the relation $$\vec{A} \cdot \vec{B} = |\vec{A} \times \vec{B}|$$. Then the value of $$|\vec{A} - \vec{B}|$$ will be:

The condition $$\vec{A} \cdot \vec{B} = |\vec{A} \times \vec{B}|$$ gives $$AB\cos\theta = AB\sin\theta$$, where $$\theta$$ is the angle between the two vectors. Dividing both sides by $$AB\cos\theta$$ yields $$\tan\theta = 1$$, so $$\theta = 45°$$.

The magnitude of the difference of two vectors is given by $$|\vec{A} - \vec{B}|^2 = A^2 + B^2 - 2\vec{A}\cdot\vec{B} = A^2 + B^2 - 2AB\cos\theta$$.

Substituting $$\theta = 45°$$, so $$\cos 45° = \frac{1}{\sqrt{2}}$$, we get $$|\vec{A} - \vec{B}|^2 = A^2 + B^2 - 2AB \cdot \frac{1}{\sqrt{2}} = A^2 + B^2 - \sqrt{2}\,AB$$.

Therefore, $$|\vec{A} - \vec{B}| = \sqrt{A^2 + B^2 - \sqrt{2}\,AB}$$.