Question #67433

Let $L_1$ and $L_2$ be two intersecting lines in 3D space with unit direction vectors $\hat{u} = l_1 \hat{i} + m_1 \hat{j} + n_1 \hat{k}$ and $\hat{v} = l_2 \hat{i} + m_2 \hat{j} + n_2 \hat{k}$ respectively. Let $\theta \in (0, \frac{\pi}{2}]$ be the acute angle between them. Let $\vec{w} = \hat{u} \times \hat{v} = w_1 \hat{i} + w_2 \hat{j} + w_3 \hat{k}$. If the following relation holds true: $\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ w_1 & w_2 & w_3 \end{vmatrix} - \begin{vmatrix} l_1 & w_1 & l_2 \\ m_1 & w_2 & m_2 \\ n_1 & w_3 & n_2 \end{vmatrix} = 1 + \cos 2\theta$ Then the value of $\theta$ is: