To determine the principal value of \( \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \), we consider that the principal value of \( \cot^{-1}(x) \) is defined within the interval \( (0, \pi) \). We seek an angle \( \theta \) such that \( \cot \theta = -\frac{1}{\sqrt{3}} \). We know that \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \). Due to the properties of the cotangent function, \( \cot \left( \pi - \frac{\pi}{3} \right) = \cot \left( \frac{2\pi}{3} \right) = -\frac{1}{\sqrt{3}} \). Since \( \frac{2\pi}{3} \) falls within the principal range \( (0, \pi) \), the principal value of \( \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \) is \( \frac{2\pi}{3} \).