Angle of Intersection of Two Lines

Let {l_1} and {l_2} be two coplanar and non-parallel lines with inclination {\alpha _1} and {\alpha _2} respectively, as shown in the given diagram. The angle of intersection of lines {l_1} and {l_2} is the angle \theta through which line {l_1} is rotated counter-clockwise about the point of intersection so that it coincides with {l_2}.


angle-bw-st-line01

The angle \theta is the angle of the intersection of lines {l_1} and {l_2} measured from {l_2} to {l_1}. The angle \psi is also the angle of intersection of lines {l_1} and {l_2} measured from {l_2} to {l_1}. If the lines are not perpendicular, then one angle between them is an acute angle.

Theorem: The angle \theta of the intersection of two non-vertical lines {l_1} and {l_2} from {l_2} to {l_1} is given by \tan \theta = \frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}, where {m_1} and {m_2} are the slopes of lines {l_1} and {l_2} respectively.


angle-bw-st-line02

Proof: Let {l_1} and {l_2} be two coplanar and non-parallel lines with inclination {\alpha _1} and {\alpha _2} respectively, as shown in the given diagram. It is clear from the diagram that

\begin{gathered} {\alpha _1} + \theta = {\alpha _2} \\ \Rightarrow \theta = {\alpha _2} - {\alpha _1} \\ \Rightarrow \tan \theta = \tan \left( {{\alpha _2} - {\alpha _1}} \right) \\ \Rightarrow \tan \theta = \frac{{\tan {\alpha _2} - \tan {\alpha _1}}}{{1 + \tan {\alpha _1}\tan {\alpha _2}}}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ \end{gathered}

Since {\alpha _1} and {\alpha _2} are the inclination of lines {l_1} and {l_2} respectively, their slopes are {m_1} = \tan {\alpha _1}, {m_2} = \tan {\alpha _2}. Putting these values of \tan {\alpha _1}, \tan {\alpha _2} in equation (i), we have

 \Rightarrow \boxed{\tan \theta = \frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}}

If the lines {l_1} and {l_2} are perpendicular, then \theta = {90^ \circ }, and using above formula, we have

\begin{gathered} 1 + {m_1}{m_2} = \frac{{{m_2} - {m_1}}}{{\tan \theta }} \\ \Rightarrow 1 + {m_1}{m_2} = \frac{{{m_2} - {m_1}}}{{\tan {{90}^ \circ }}} \\ \Rightarrow 1 + {m_1}{m_2} = \frac{{{m_2} - {m_1}}}{\infty }\,\,\, \Rightarrow 1 + {m_1}{m_2} = 0 \\ \end{gathered}


\boxed{{m_1}{m_2} = - 1}

This is the condition for two lines to be perpendicular.