Understanding Line Intersection in 3D
In 3D space, two lines can intersect (meet at a single point), be parallel (no intersection), or be skew (not parallel and never meet).
We present a calculator and a step by step solver to find points of intersection of two lines in 3D if any.
Line through two points:
$$L_1: \vec{r_1}(t) = \langle x_A, y_A, z_A \rangle + t\,\langle x_B-x_A, y_B-y_A, z_B-z_A \rangle$$
$$L_2: \vec{r_2}(s) = \langle x_C, y_C, z_C \rangle + s\,\langle x_D-x_C, y_D-y_C, z_D-z_C \rangle$$
Parametric form:
$$L_1: \begin{cases} x = x_1 + t\,a_1 \\ y = y_1 + t\,b_1 \\ z = z_1 + t\,c_1 \end{cases}$$
$$L_2: \begin{cases} x = x_2 + s\,a_2 \\ y = y_2 + s\,b_2 \\ z = z_2 + s\,c_2 \end{cases}$$
To find the intersection: Solve the system:
$$\begin{cases}
x_A + t\,(x_B-x_A) = x_C + s\,(x_D-x_C) \\
y_A + t\,(y_B-y_A) = y_C + s\,(y_D-y_C) \\
z_A + t\,(z_B-z_A) = z_C + s\,(z_D-z_C)
\end{cases}$$
Important: Solve two equations for t and s, then verify if they satisfy the third equation. If the third equation is satisfied, the lines intersect. If not, they are skew. If the direction vectors are parallel, lines are either parallel or coincident.