Angle Between Two Lines

Definition. The angle between two lines in space, with direction vectors $\vec b_1$ and $\vec b_2$, is defined as the acute (or right) angle $\theta$ between their directions -- taking whichever of the two supplementary angles formed is $\le90^\circ$, since a direction vector for a line could equally well be taken in either of its two opposite senses ($\vec b_1$ or $-\vec b_1$); reporting an obtuse angle would make the answer depend on an arbitrary sign choice, so the convention is fixed by always taking the non-negative acute value.

Formula via the dot product -- derivation. For any two vectors, the dot product satisfies $\vec b_1\cdot\vec b_2=|\vec b_1||\vec b_2|\cos\theta_0$, where $\theta_0$ is the actual angle between them as directed vectors (which could be obtuse). Solving for $\cos\theta_0$ and taking the absolute value to enforce the acute-angle convention gives the angle between two lines:

Written out for direction ratios $a_1,b_1,c_1$ and $a_2,b_2,c_2$ (expanding the dot product and magnitudes into coordinates), this is

If the two lines are instead given by their direction cosines $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$ directly, the magnitudes in the denominator are each exactly $1$ (Section 4), so the formula simplifies to

Special cases. Two important conditions follow immediately, and are used constantly without re-deriving the angle each time:

• Parallel lines ($\theta=0$, so $\cos\theta=1$): this happens exactly when $\vec b_1$ and $\vec b_2$ are scalar multiples of each other, i.e. $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$.
• Perpendicular lines ($\theta=90^\circ$, so $\cos\theta=0$): this happens exactly when the dot product vanishes, $a_1a_2+b_1b_2+c_1c_2=0$ (equivalently $l_1l_2+m_1m_2+n_1n_2=0$) -- a simple linear condition, used to find an unknown constant in one line's direction ratios that makes it perpendicular to another given line. …