Condition for Two Lines to be Intersecting, Coplanar and the Plane Containing Them (Cartesian Form)

In 3D geometry, understanding the relationship between two lines is crucial. Specifically, we are interested in when two lines intersect, when they are coplanar, and how to determine the plane that contains them. Let's explore these concepts in detail.

Intersecting Lines

Two lines in space will intersect if they cross each other at a single point. For two lines to intersect, they must lie in the same plane (be coplanar) and cannot be parallel.

Condition for Intersection

In Cartesian form, two lines are represented by their parametric equations:

$$ \begin{align*} \text{Line } L_1 &: \frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1} \ \text{Line } L_2 &: \frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2} \end{align*} $$

For these lines to intersect, there must exist values of the parameters (let's call them $t_1$ for $L_1$ and $t_2$ for $L_2$) such that:

$$ \begin{align*} x_1 + a_1 t_1 &= x_2 + a_2 t_2 \ y_1 + b_1 t_1 &= y_2 + b_2 t_2 \ z_1 + c_1 t_1 &= z_2 + c_2 t_2 \end{align*} $$

If such $t_1$ and $t_2$ exist, the lines intersect at the point $(x_1 + a_1 t_1, y_1 + b_1 t_1, z_1 + c_1 t_1)$.

Coplanar Lines

Two lines are coplanar if they lie in the same plane. This is true whether the lines intersect or are parallel.

Condition for Coplanarity

For two lines to be coplanar, the vector connecting a point on one line to a point on the other line must be coplanar with the direction vectors of the two lines. The condition for coplanarity can be expressed using the scalar triple product:

$$ \begin{vmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \end{vmatrix} = 0 $$

If the scalar triple product is zero, the lines are coplanar.

Plane Containing Two Lines

If two lines are coplanar, there is a unique plane that contains both lines. The equation of the plane can be found using a point from one of the lines and the cross product of the direction vectors of the two lines.

Equation of the Plane

Let's denote the cross product of the direction vectors of $L_1$ and $L_2$ as $\vec{n} = \vec{d_1} \times \vec{d_2}$, where $\vec{d_1} = \langle a_1, b_1, c_1 \rangle$ and $\vec{d_2} = \langle a_2, b_2, c_2 \rangle$. The equation of the plane containing the two lines is:

$$ \vec{n} \cdot (\vec{r} - \vec{r_0}) = 0 $$

where $\vec{r} = \langle x, y, z \rangle$ is a general point on the plane, and $\vec{r_0} = \langle x_1, y_1, z_1 \rangle$ is a point on one of the lines (for example, $L_1$).

Table of Differences and Important Points

Property	Intersecting Lines	Coplanar Lines	Plane Containing Lines
Definition	Cross at a single point	Lie in the same plane	A flat surface containing both lines
Condition	Existence of parameters $t_1$ and $t_2$	Scalar triple product is zero	Cross product of direction vectors
Equation	Solve system of equations for $t_1$ and $t_2$	$\begin{vmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \end{vmatrix} = 0$	$\vec{n} \cdot (\vec{r} - \vec{r_0}) = 0$

Examples

Example 1: Intersecting Lines

Consider two lines:

$$ L_1: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{4}, \quad L_2: \frac{x - 3}{-1} = \frac{y - 2}{5} = \frac{z + 1}{2} $$

To find if they intersect, we solve the system of equations for $t_1$ and $t_2$. If a solution exists, the lines intersect.

Example 2: Coplanar Lines

For the same lines $L_1$ and $L_2$, we check the scalar triple product:

$$ \begin{vmatrix} 2 & 3 & 4 \ -1 & 5 & 2 \ 3 - 1 & 2 + 1 & 1 - 2 \end{vmatrix} = 0 $$

If this determinant is zero, the lines are coplanar.

Example 3: Plane Containing Lines

If $L_1$ and $L_2$ are coplanar, we find the cross product of their direction vectors:

$$ \vec{n} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 3 & 4 \ -1 & 5 & 2 \end{vmatrix} $$

Using a point from $L_1$, say $(1, -1, 2)$, we can write the equation of the plane:

$$ \vec{n} \cdot (x - 1, y + 1, z - 2) = 0 $$

This equation represents the plane containing both $L_1$ and $L_2$.