Intersections of a Line with a Plane

Introduction

The intersection of a line and a plane is a fundamental concept in 3D geometry. Determining the point of intersection (if it exists) involves solving equations derived from their respective representations. This topic also extends to analyzing the relationships between two or more planes and the angles between geometric objects in space.

Intersection of a Line and a Plane

The general vector equation of a line is:

\[ \vec{r} = \vec{a} + \lambda \vec{b} \]

Here:

• \( \vec{r} \): Position vector of any point on the line.
• \( \vec{a} \): Position vector of a known point on the line.
• \( \vec{b} \): Direction vector of the line.
• \( \lambda \): Scalar parameter.

The vector equation of a plane is:

\[ \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} \]

or equivalently in Cartesian form:

\[ ax + by + cz = d \]

Here:

• \( \vec{n} \): Normal vector to the plane.
• \( a, b, c \): Components of \( \vec{n} \).
• \( d \): A constant representing \( \vec{a} \cdot \vec{n} \), where \( \vec{a} \) is a known point on the plane.

Finding the Intersection

To find the point of intersection of the line and plane:

1. Substitute the parametric equation of the line \( \vec{r} = \vec{a} + \lambda \vec{b} \) into the equation of the plane \( ax + by + cz = d \).
2. Solve for the parameter \( \lambda \).
3. Use \( \lambda \) to calculate the coordinates of the intersection point.

Geometric Interpretation

• If a solution exists, the line intersects the plane at a single point.
• If no solution exists, the line is parallel to the plane.
• If the line lies entirely within the plane, the equations will be dependent.

Intersection of Two or More Planes

• Two Planes: The intersection of two planes is either a line (if they are not parallel) or they do not intersect (if they are parallel).
• Three Planes: Three planes may intersect at a point, form a line, or have no intersection (depending on their orientations).

Solving Intersections

To find intersections involving planes, solve the system of linear equations derived from their Cartesian forms:

\[ a_1x + b_1y + c_1z = d_1 \]

\[ a_2x + b_2y + c_2z = d_2 \]

\[ a_3x + b_3y + c_3z = d_3 \]

Angle Between a Line and a Plane

The angle \( \theta \) between a line (direction vector \( \vec{b} \)) and a plane (normal vector \( \vec{n} \)) is given by:

\[ \sin \theta = \frac{|\vec{b} \cdot \vec{n}|}{|\vec{b}| |\vec{n}|} \]

Angle Between Two Planes

The angle \( \phi \) between two planes is determined by the angle between their normal vectors \( \vec{n_1} \) and \( \vec{n_2} \):

\[ \cos \phi = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}| |\vec{n_2}|} \]

Examples

Example 1: Intersection of a Line and a Plane

Given the line \( \vec{r} = [1, 2, 3] + \lambda [2, 1, -1] \) and the plane \( 2x + y – z = 5 \):

1. 1. Substitute \( x = 1 + 2\lambda \), \( y = 2 + \lambda \), \( z = 3 – \lambda \) into \( 2x + y – z = 5 \):

\[ 2(1 + 2\lambda) + (2 + \lambda) – (3 – \lambda) = 5 \]

1. Simplify to find \( \lambda = 1 \).
2. Substitute \( \lambda = 1 \) into the parametric equations to find the intersection point \( [3, 3, 2] \).

Example 2: Angle Between a Line and a Plane

For a line with direction vector \( \vec{b} = [2, 1, -1] \) and a plane with normal vector \( \vec{n} = [2, 1, 1] \):

\[ \sin \theta = \frac{|(2)(2) + (1)(1) + (-1)(1)|}{\sqrt{(2^2 + 1^2 + (-1)^2)} \sqrt{(2^2 + 1^2 + 1^2)}} \]

\[ \sin \theta = \frac{|4 + 1 – 1|}{\sqrt{6} \sqrt{6}} = \frac{4}{6} \]

\[ \theta = \arcsin\left(\frac{2}{3}\right) \]

Applications

• Engineering: Designing intersections in 3D structures.
• Physics: Analyzing forces and trajectories in space.
• Computer Graphics: Rendering and collision detection.