Equation of Straight Line in Plane/Two-Point Form
Theorem
Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.
Let $\LL$ be the straight line passing through $p_1$ and $p_2$.
Then $\LL$ can be described by the equation:
- $\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
or:
- $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
Parametric Form
Let $\LL$ be a straight line embedded in a cartesian plane, given in two-point form as:
- $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
Then $\LL$ can be expressed by the parametric equations:
- $\begin {cases} x = x_1 + t \paren {x_2 - x_1} \\ y = y_1 + t \paren {y_2 - y_1} \end {cases}$
Determinant Form
Let $\LL$ be a straight line embedded in a cartesian plane, given in two-point form as:
- $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
Then $\LL$ can be expressed in the form:
- $\begin {vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end {vmatrix}$
Proof
From the slope-intercept form of the equation of the straight line:
- $(1): \quad y = m x + c$
which is to be satisfied by both $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.
We express $m$ and $c$ in terms of $\paren {x_1, y_1}$ and $\paren {x_2, y_2}$:
\(\ds y_1\) | \(=\) | \(\ds m x_1 + c\) | ||||||||||||
\(\ds y_2\) | \(=\) | \(\ds m x_2 + c\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds c\) | \(=\) | \(\ds y_1 - m x_1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y_2\) | \(=\) | \(\ds m x_2 + y_1 - m x_1\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds m\) | \(=\) | \(\ds \dfrac {y_2 - y_1} {x_2 - x_1}\) |
\(\ds y_1\) | \(=\) | \(\ds m x_1 + c\) | ||||||||||||
\(\ds y_2\) | \(=\) | \(\ds m x_2 + c\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds m\) | \(=\) | \(\ds \dfrac {y_2 - c} {x_2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y_1\) | \(=\) | \(\ds \dfrac {y_2 - c} {x_2} x_1 + c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y_1 x_2\) | \(=\) | \(\ds x_1 y_2 + c \paren {x_2 - x_1}\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds c\) | \(=\) | \(\ds \dfrac {y_1 x_2 - x_1 y_2} {x_2 - x_1}\) |
Substituting for $m$ and $c$ in $(1)$:
\(\ds y\) | \(=\) | \(\ds m x + c\) | which is $(1)$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \dfrac {y_2 - y_1} {x_2 - x_1} x + \dfrac {y_1 x_2 - x_1 y_2} {x_2 - x_1}\) | from $(2)$ and $(3)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y \paren {x_2 - x_1} + x_1 y_2\) | \(=\) | \(\ds x \paren {y_2 - y_1} + y_1 x_2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y \paren {x_2 - x_1} + x_1 y_2 - y_1 x_1\) | \(=\) | \(\ds x \paren {y_2 - y_1} + y_1 x_2 - x_1 y_1\) | adding $y_1 x_1 = x_1 y_1$ to both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y \paren {x_2 - x_1} - y_1 \paren {x_2 - x_1}\) | \(=\) | \(\ds x \paren {y_2 - y_1} - x_1 \paren {y_2 - y_1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {y - y_1} \paren {x_2 - x_1}\) | \(=\) | \(\ds \paren {x - x_1} \paren {y_2 - y_1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {y - y_1} {x - x_1}\) | \(=\) | \(\ds \dfrac {y_2 - y_1} {x_2 - x_1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {x - x_1} {x_2 - x_1}\) | \(=\) | \(\ds \dfrac {y - y_1} {y_2 - y_1}\) |
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $4$. Special forms of the equation of a straight line: $(3)$ Line through two points
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: $10.3$: Equation of Line joining Two Points $\map {P_1} {x_1, y_1}$ and $\map {P_2} {x_2, y_2}$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $11$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: line: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: line: 2.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: line (in two dimensions) Two-point form
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: straight line (in the plane)