Calculation of the Intersection Point of a Line and a Plane in Space
Introduction
In three-dimensional space, the intersection of a line and a plane is a fundamental concept in geometry and linear algebra. The intersection point represents the unique location where the line and the plane meet, provided they are not parallel. This analysis explores the mathematical framework for determining such a point, emphasizing the role of parametric equations and vector algebra. The study also addresses the conditions under which the line and plane intersect, including scenarios where they are parallel or coincident.
Mathematical Formulation
To calculate the intersection point of a line and a plane, we begin by defining the line and the plane using their respective equations. A line in 3D space can be represented parametrically as:
$$
\mathbf{r}(t) = \mathbf{a} + t\mathbf{v}
$$
where $\mathbf{a}$ is a point on the line, $\mathbf{v}$ is the direction vector of the line, and $t$ is a scalar parameter. A plane in 3D space is typically expressed by the equation:
$$
Ax + By + Cz + D = 0
$$
where $(A, B, C)$ is the normal vector of the plane. The intersection of the line and the plane occurs when the parametric equations of the line satisfy the plane's equation for some value of $t$.
To find this point, substitute the parametric expressions of the line into the plane's equation. For instance, if the line is given by:
$$
x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct
$$
and the plane is given by:
$$
Ax + By + Cz + D = 0
$$
substituting the line's equations into the plane's equation results in an equation in $t$. Solving this equation for $t$ yields the parameter value at which the line intersects the plane. Once $t$ is determined, the corresponding coordinates of the intersection point can be calculated by substituting $t$ back into the line's parametric equations.
Conditions for Intersection
The existence of an intersection point between a line and a plane depends on their relative orientations.
- Parallelism: If the direction vector $\mathbf{v}$ of the line is perpendicular to the normal vector $(A, B, C)$ of the plane, the line is parallel to the plane. In this case, the line does not intersect the plane unless it is coincident with it.
- Coincidence: If the line lies entirely within the plane (i.e., all points of the line satisfy the plane's equation), the line and plane are coincident, resulting in an infinite number of intersection points.
- Single Intersection Point: When the line is not parallel to the plane and is not coincident with it, the line and plane intersect at exactly one point. This occurs when the system of equations derived from substituting the line's parametric equations into the plane's equation has a unique solution for $t$.
Example Calculation
Consider a line defined by the parametric equations:
$$
x = 1 + 2t, \quad y = 2 + 3t, \quad z = 4 + t
$$
and a plane defined by the equation:
$$
2x + 3y - z = 6
$$
Substituting the line's equations into the plane's equation:
$$
2(1 + 2t) + 3(2 + 3t) - (4 + t) = 6
$$
Simplify:
$$
2 + 4t + 6 + 9t - 4 - t = 6
\Rightarrow 4 + 12t = 6
\Rightarrow 12t = 2
\Rightarrow t = \frac{1}{6}
$$
Substituting $t = \frac{1}{6}$ back into the line's equations:
$$