Free Intersection of Two Lines Calculator

Discovering the Point of Intersection of Two Lines

The Intersection of Two Lines Calculator is an online tool designed to help you quickly find the intersection point of two lines, whether you're working in 2D or 3D. It accepts common equation formats—slope-intercept and general (standard) form for 2D, and parametric equations for 3D—making it easy to handle line intersection problems without manual algebra.

Lines That Meet: Understanding Intersections

Intersecting lines are lines that cross each other at exactly one location, known as the point of intersection. If two lines share more than one point, they coincide (are identical). In a plane, lines that never meet are parallel. In 3D space, a third possibility exists: skew lines. Skew lines are not parallel yet still do not intersect because they lie in different planes. The 2D line intersection scenario only allows crossing or parallel; 3D line intersection adds the skew case.

2D Intersection Formulas

The calculator supports two common representations.

Slope-Intercept Form

Given lines $y = m_1 x + b_1$ and $y = m_2 x + b_2$, the intersection coordinates $(x_0, y_0)$ are:

$$x_0 = \dfrac{b_2 - b_1}{m_1 - m_2}, \qquad y_0 = m_1 \dfrac{b_2 - b_1}{m_1 - m_2} + b_1$$

This formula works as long as the slopes differ ($m_1 \neq m_2$). If the slopes are equal, the lines are either parallel or coincident, and no unique intersection exists.

General (Standard) Form

For lines $A_1 x + B_1 y = C_1$ and $A_2 x + B_2 y = C_2$, the line intersection point can be found using:

$$x_0 = \dfrac{B_2 C_1 - B_1 C_2}{A_1 B_2 - A_2 B_1}, \qquad y_0 = \dfrac{A_1 C_2 - A_2 C_1}{A_1 B_2 - A_2 B_1}$$

The denominator $A_1 B_2 - A_2 B_1$ becomes zero for parallel or coincident lines, indicating no single crossing point.

Lines in 3D: Parametric Approach

In three dimensions, lines are typically expressed parametrically:

$$\text{Line 1: } \begin{cases} x = p_1 + t \, d_1 \\ y = p_2 + t \, d_2 \\ z = p_3 + t \, d_3 \end{cases}, \qquad \text{Line 2: } \begin{cases} x = q_1 + s \, e_1 \\ y = q_2 + s \, e_2 \\ z = q_3 + s \, e_3 \end{cases}$$

Here $t$ and $s$ are real parameters. To find the intersection of two lines in 3D, we set the coordinates equal and solve for $t$ and $s$. If a unique solution $(t_0, s_0)$ exists, the intersection point is obtained by substituting $t_0$ into Line 1 (or $s_0$ into Line 2).

Worked Example

Let's find the intersection point of these two lines:

• L1: $x = 2 + t,\; y = 3 + 2t,\; z = 4 + 3t$
• L2: $x = 3,\; y = 3 + s,\; z = 5 + s$

We equate the coordinates:

$$\begin{aligned} 2 + t &= 3 \quad &\Rightarrow\quad t &= 1 \\ 3 + 2t &= 3 + s \quad &\Rightarrow\quad 3 + 2 &= 3 + s \;\Rightarrow\; s = 2 \\ 4 + 3t &= 5 + s \quad &\Rightarrow\quad 4 + 3 &= 5 + 2 \;\Rightarrow\; 7 = 7 \quad \text{(consistent)} \end{aligned}$$

Thus $t_0 = 1$, $s_0 = 2$. Substituting into L1 gives the point of intersection:

$$(x_0, y_0, z_0) = (3,\;5,\;7)$$

You can verify this by plugging $s = 2$ into L2: $(3,\;5,\;7)$ as well.

Using the Calculator

To use the tool, specify whether your problem is 2D or 3D. For 2D, choose between slope-intercept and general form; for 3D, enter the parametric equations of both lines. The calculator instantly returns the intersection coordinates or indicates that the lines do not intersect (parallel, coincident, or skew). The slope-intercept intersection and parametric line intersection modes make this calculator versatile for anyone studying geometry or working in design, engineering, or mathematics.