The angle between two lines with direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ satisfies:

Each set of direction cosines $(l, m, n)$ corresponds to a unit vector along the line. The angle $\theta$ between two unit vectors is the dot product: $\cos\theta = l_1 l_2 + m_1 m_2 + n_1 n_2$. Consequence: lines are perpendicular iff $l_1 l_2 + m_1 m_2 + n_1 n_2 = 0$. Lines are parallel iff direction cosines are proportional (same up to sign).