Introduction

Finding the line of intersection between two planes is a common problem in vector calculus and geometry. It has applications in computer graphics, engineering, and three-dimensional modeling. This article will provide a thorough explanation of the mathematical process involved, supported by a clear example. We'll also include a table summarizing the key points and formulae involved in the process.

Theoretical Background

Equation of a Plane

A plane in three-dimensional space can be defined by a linear equation of the form:

$$ax + by + cz = d$$

where $(a, b, c)$ is the normal vector to the plane, and $d$ is the perpendicular distance from the origin to the plane along the normal.

Intersection of Two Planes

Let's consider two planes defined by the equations:

1. $a_1x + b_1y + c_1z = d_1$
2. $a_2x + b_2y + c_2z = d_2$

The intersection of these two planes is a line if they are not parallel. When the planes intersect, their normal vectors $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are not scalar multiples of each other.

Determining the Line of Intersection

To find the line of intersection of two planes:

1. Calculate the Direction Vector: The cross product of the normal vectors of the two planes will give the direction vector of the line of intersection. Symbolically, it is:
   $$\mathbf{d} = (a_1, b_1, c_1) \times (a_2, b_2, c_2)$$
   This vector will lie in both planes.
2. Find a Specific Point on the Line: To find a point through which the line passes, solve the system of equations given by the two plane equations. This system can be solved using substitution or elimination by treating one variable in terms of the others or setting one variable equal to a constant value. The solution will be the coordinates of a point on the line.
3. Equation of the Line: Using the direction vector $\mathbf{d}$ and a specific point $\mathbf{p_0}$, the parametric equation for the line is given by:
   $$\mathbf{r}(t) = \mathbf{p_0} + t\mathbf{d}$$
   Where $t$ is a parameter.

Example

Consider the intersection of the planes:

1. $x + 2y + 3z = 4$
2. $2x + y + z = 1$

Step 1: Calculate the Direction Vector

• The normal vectors are $(1, 2, 3)$ and $(2, 1, 1)$. • The cross product is:

$$\mathbf{d} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 3 \\ 2 & 1 & 1 \\ \end{vmatrix} = (1*1 - 3*1)\mathbf{i} - (1*1 - 3*2)\mathbf{j} + (1*1 - 2*2)\mathbf{k} = (-2, 5, -3)$$

Step 2: Find a Specific Point on the Line

• Set $z = 0$ to simplify calculations. Then solve:

• From Plane 1: $x + 2y = 4$ • From Plane 2: $2x + y = 1$

Solving these:

1. $x = 4 - 2y$
2. Substituting into Plane 2: $2(4 - 2y) + y = 1 \Rightarrow 8 - 4y + y = 1 \Rightarrow 3y = 7 \Rightarrow y = \frac{7}{3}$
3. $x = 4 - 2 \times \frac{7}{3} = \frac{4}{3}$

• Hence, a specific point $\mathbf{p_0}$ on the line is $\left(\frac{4}{3}, \frac{7}{3}, 0\right)$.

Step 3: Equation of the Line

• The parametric form is:

$$\mathbf{r}(t) = \left(\frac{4}{3}, \frac{7}{3}, 0\right) + t(-2, 5, -3)$$

• This can be expressed in parametric equations as:

$$x(t) = \frac{4}{3} - 2t$$ $$y(t) = \frac{7}{3} + 5t$$ $$z(t) = -3t$$

Summary Table

Additional Considerations

• Parallel and Coincident Planes: If the directional vector is the zero vector, the planes are either parallel or coincident.

• Applications: This calculation is particularly useful in 3D computer graphics where determining line intersections is crucial for rendering and computational geometry algorithms.

This detailed procedure serves as a comprehensive guide to understanding and implementing the line intersection of two planes, enhancing both mathematical understanding and practical application skills.