Angle Between Two Lines (Direction Cosines)

Acute angle θ between two lines with direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂). The absolute value ensures the acute angle is taken.

Let two lines have direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$. Their unit direction vectors are $\hat{u}_1 = (l_1, m_1, n_1)$ and $\hat{u}_2 = (l_2, m_2, n_2)$.

The angle $\theta$ between the lines satisfies:

$$\cos\theta = \hat{u}_1 \cdot \hat{u}_2 = l_1l_2+m_1m_2+n_1n_2$$

Since each vector is a unit vector, $|\hat{u}_1||\hat{u}_2| = 1$, so no normalization is needed.

Taking the acute angle: $\cos\theta = |l_1l_2+m_1m_2+n_1n_2|$.

Perpendicularity: $l_1l_2+m_1m_2+n_1n_2 = 0$.

Parallelism: $l_1 = l_2$, $m_1 = m_2$, $n_1 = n_2$ (same direction) or all negatives (opposite direction).

Note on angle between skew lines: Two lines in 3D that do not intersect (skew lines) still have a well-defined angle — it is the angle between their direction vectors, as if translated to meet at a point.