Intersection of Straight Lines in General Form
Theorem
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, given by the equations:
| \(\ds \LL_1: \ \ \) | \(\ds l_1 x + m_1 y + n_1\) | \(=\) | \(\ds 0\) | |||||||||||
| \(\ds \LL_2: \ \ \) | \(\ds l_2 x + m_2 y + n_2\) | \(=\) | \(\ds 0\) |
The point of intersection of $\LL_1$ and $\LL_2$ has coordinates given by:
- $\dfrac x {m_1 n_2 - m_2 n_1} = \dfrac y {n_1 l_2 - n_2 l_1} = \dfrac 1 {l_1 m_2 - l_2 m_1}$
This point exists and is unique if and only if $l_1 m_2 \ne l_2 m_1$.
Determinant Form
This result can also be expressed in the form:
- $\dfrac x {\begin {vmatrix} m_1 & n_1 \\ m_2 & n_2 \end {vmatrix} } = \dfrac y {\begin {vmatrix} n_1 & l_1 \\ n_2 & l_2 \end {vmatrix} } = \dfrac 1 {\begin {vmatrix} l_1 & m_1 \\ l_2 & m_2 \end {vmatrix} }$
where $\begin {vmatrix} \cdot \end {vmatrix}$ denotes a determinant.
Proof
First note that by the parallel postulate $\LL_1$ and $\LL_2$ have a unique point of intersection if and only if they are not parallel.
From Condition for Straight Lines in Plane to be Parallel, $\LL_1$ and $\LL_2$ are parallel if and only if $l_1 m_2 = l_2 m_1$.
$\Box$
Let the equations for $\LL_1$ and $\LL_2$ be given.
Let $\tuple {x, y}$ be the point on both $\LL_1$ and $\LL_2$.
We have:
| \(\ds l_1 x + m_1 y + n_1\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -\dfrac {l_1} {m_1} x - \dfrac {n_1} {m_1}\) | |||||||||||
| \(\ds l_2 x + m_2 y + n_2\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -\dfrac {l_2} {m_2} x - \dfrac {n_2} {m_2}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {l_1} {m_1} x + \dfrac {n_1} {m_1}\) | \(=\) | \(\ds \dfrac {l_2} {m_2} x + \dfrac {n_2} {m_2}\) | substituting for $y$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds l_1 m_2 x + n_1 m_2\) | \(=\) | \(\ds l_2 m_1 x + n_2 m_1\) | multiplying by $m_1 m_2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x \paren {l_1 m_2 - l_2 m_1}\) | \(=\) | \(\ds n_2 m_1 - n_1 m_2\) | rearranging | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac x {m_1 n_2 - m_2 n_1}\) | \(=\) | \(\ds \dfrac 1 {l_1 m_2 - l_2 m_1}\) | dividing by $\paren {l_1 m_2 - l_2 m_1} \paren {m_1 n_2 - m_2 n_1}$ |
Similarly:
| \(\ds l_1 x + m_1 y + n_1\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds -\dfrac {m_1} {l_1} y - \dfrac {n_1} {l_1}\) | |||||||||||
| \(\ds l_2 x + m_2 y + n_2\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds -\dfrac {m_2} {l_2} y - \dfrac {n_2} {l_2}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {m_1} {l_1} y + \dfrac {n_1} {l_1}\) | \(=\) | \(\ds \dfrac {m_2} {l_2} y + \dfrac {n_2} {l_2}\) | substituting for $x$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds m_1 l_2 y + n_1 l_2\) | \(=\) | \(\ds m_2 l_1 y + n_2 l_1\) | multiplying by $m_1 m_2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y \paren {m_2 l_1 - m_1 l_2}\) | \(=\) | \(\ds n_1 l_2 - n_2 l_1\) | rearranging | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac y {n_1 l_2 - n_2 l_1}\) | \(=\) | \(\ds \dfrac 1 {l_1 m_2 - l_2 m_1}\) | dividing by $\paren {l_1 m_2 - l_2 m_1} \paren {n_1 l_2 - n_2 l_1}$ |
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $8$. Intersection of two lines