Contents

Geometry plays a crucial role in our understanding of the physical world. In advanced geometry, particularly in three-dimensional space, **planes** are infinite flat surfaces that extend indefinitely in all directions.

**How to find the intersection of two planes?** This article delves into the methods of finding the **intersection of two planes** and the significance of the** intersection line**.

## How to Find the Intersection of Two Planes

To find the** intersection of two planes**, we solve their equations simultaneously. The intersection can be a line (given as a point and a direction) or can be the entire plane (if they’re identical) or no intersection (if they’re parallel).

To further understand the concept, we need to delve into the understanding of planes first, let us begin with some fundamentals of a plane.

**Introduction to Planes**

In **three-dimensional space**, a plane can be defined using a point and a normal vector to the plane or using three non-collinear points. The equation of a plane is generally represented in the form:

$ax+by+cz=d$

Where $a,b,c$ are the coefficients of the normal vector to the plane and $d$ is a constant.

Figure-1.

**Understanding Intersections**

When **two planes intersect**, they typically do so along a straight line, unless they are parallel. This** intersection line** will have its unique direction vector, determined by the cross product of the normal vectors of the** two planes**.

However, if **two planes** are parallel and distinct, they won’t intersect. If they are coincident (i.e., the same plane), then their **intersection is the plane** itself.

**Finding the Intersection of Two Planes**

**Determine the Direction Vector of the Line of Intersection**- Begin by identifying the normal vectors of the
**two planes.**If your**planes**are defined by the equations:

- Begin by identifying the normal vectors of the

$a_{1 *}x+b_{1 *}y+c_{1 *}z=d_{1}$

$a_{2}x+b_{2 *}y+c_{2 *}z=d_{2}$

The normal vectors are $n_{1}= <a_{1},b_{1},c_{1}>$ and $<a_{2},b_{2},c_{2}>$.

- Calculate the direction vector of the
**intersection line**$d$, by taking the cross product of $n_{1}$ and $n_{2}$.

- Calculate the direction vector of the
**Find a Point on the Line of Intersection**

One approach is to set one of the variables (x, y, or z) to a constant value, then solve the two plane equations simultaneously to get the values of the other two variables. This will give you a point on the **intersection line**.

**Determine the Parametric Equations**

With the direction vector $d$ and a point $P(x_{0},y_{0},z_{0})$ on the line, the parametric equations of the **intersection line** can be expressed as:

$x=x_{0}+_{*}t_{1}$

$y=y_{0}+_{*}t_{2}$

$z=z_{0}+_{*}t_{3}$

Where $t$ is a parameter and $(t_{1},t_{2},t_{3})$ are the components of the direction vector $d$.

## Exercise

**Example 1**

**Plane 1**: $x+y+z=6$**Plane 2**: $x−y+z=2$

### Solution

To find their **intersection**, equate z from both equations:

From **Plane 1**: $z=6−x−y$

Substitute in** Plane 2**: $x−y+6−x−y=2$

This gives: $−2y=−4$

Thus, $y=2$

Now, using $y=2$ in Plane 1: $x+2+z=6$ => $x+z=4$

So, the **intersection line** is: $x+z=4$, and $y=2$.

**Example 2**

**Plane 1**: $2x−3y+z=5$**Plane 2**: $x+y−z=1$

Figure-2.

### Solution

Equating z from both:

From **Plane 1**: $z=5−2x+3y$

Substitute in **Plane 2**: $x+y−5+2x−3y=1$

This yields: $3x−2y=6$.

Using **Plane 1** for z: $z=5−2x+3y$**Intersection line** is described by: $3x−2y=6$, and $z=5−2x+3y$.

**Example 3**

**Plane 1**: $3x−y−2z=7$**Plane 2**: $x+2y−z=2$

### Solution

From **Plane 1**: $z= (x−y− $

Substitute in **Plane 2**: $x+2y− (x−y− =2$

This results in: $y=−x+3$

Using **Plane 1** for z: $z= (x−y−$**Intersection line**: $y=−x+3$, and $z= (x−y−$

**Example 4**

**Plane 1**: $x−y+z=2$**Plane 2**: $2x+y+3z=6$

### Solution

From **Plane 1**: $z=2−x+y$

Substitute in **Plane 2**: $2x+y+3(2−x+y)=6$

This simplifies to: $x+4y=0$

Using **Plane 1** for z: $z=2−x+y$

**Intersection line**: $x+4y=0$, and $z=2−x+y$.

**Example 5**

**Plane 1**: $3x+y+z=4$**Plane 2**: $x−2y−z=1$

Figure-3.

### Solution

From **Plane 1**: $z=4−3x−y$

Substitute into **Plane 2**: $x−2y−4+3x+y=1$

This gives: $4x−y=5$

Using **Plane 1** for z: $z=4−3x−y$

**Intersection** line: $4x−y=5$, and $z=4−3x−y$.

**Real-World Implications of Finding the Intersection of Two Planes**

The mathematical principle of determining the **intersection of two planes** might seem abstract, but its real-world applications abound across various fields. Unraveling where two **planes** meet is essential in numerous sectors, be it technology, engineering, design, or even natural phenomena. Here’s a look at how this geometric concept integrates seamlessly into our daily lives.

**Civil Engineering and Architecture**- In the design and construction of infrastructure, understanding where
**two planes intersect**helps in determining stress points, especially in complex structures. - For architects,
**intersections**can form design features, such as where the roof meets a wall, or how multi-level terraces might interact with each other.

- In the design and construction of infrastructure, understanding where
**Computer Graphics and Animation**- Rendering realistic environments often requires determining where surfaces meet, which is essentially calculating the
**intersection of planes**. Shadows, reflections, and even character interactions might be based on these calculations. - Gaming environments, where characters navigate complex terrains, interact with objects, or when objects crash into each other, rely on
**intersection**calculations.

- Rendering realistic environments often requires determining where surfaces meet, which is essentially calculating the
**Air Traffic Control**- Air traffic controllers ensure that aircraft maintain safe distances from each other. In a three-dimensional airspace, flight paths can be considered as
**planes.**Recognizing potential**intersections**helps in preventing mid-air collisions.

- Air traffic controllers ensure that aircraft maintain safe distances from each other. In a three-dimensional airspace, flight paths can be considered as
**Geology and Geography**- Studying the Earth’s layers, especially during phenomena like plate tectonics, can be understood in terms of plane
**intersections**. For instance, where two tectonic plates meet (converge), mountain ranges or deep ocean trenches can form. - In topographical mapping, the
**intersection of planes**can represent ridgelines or river valleys.

- Studying the Earth’s layers, especially during phenomena like plate tectonics, can be understood in terms of plane
**Astronomy**- The study of celestial mechanics, especially when plotting the orbits of planets and other celestial bodies, can involve understanding the points where orbits (considered as
**planes**in space) intersect. - Phenomena like eclipses occur when the
**planes**of orbits of the Earth and the Moon intersect, casting shadows on each other.

- The study of celestial mechanics, especially when plotting the orbits of planets and other celestial bodies, can involve understanding the points where orbits (considered as
**Medicine**- In medical imaging, like MRI or CT scans, images are taken in slices or
**planes.**To understand a three-dimensional structure or locate a tumor, radiologists might look at the**intersections of these planes**.

- In medical imaging, like MRI or CT scans, images are taken in slices or
**Design and Art**- Sculptors, especially those working with materials like stone or metal, need to understand the
**intersection of planes**to shape their artwork correctly. - In graphic design, understanding how
**two planes interact**can aid in creating depth, perspective, and layers in an artwork.

- Sculptors, especially those working with materials like stone or metal, need to understand the
**Marine Navigation**- Just like in air traffic, ship navigators need to be aware of potential
**intersections**with other vessels, especially in busy shipping lanes or narrow straits. This becomes even more crucial in foggy conditions or at night.

- Just like in air traffic, ship navigators need to be aware of potential
**Meteorology**- The study of weather patterns, particularly fronts (where warm and cold air masses meet), can be conceptualized as the
**intersection of two planes**. These**intersections**often result in weather events like thunderstorms or shifts in wind patterns.

- The study of weather patterns, particularly fronts (where warm and cold air masses meet), can be conceptualized as the
**Economics and Market Research**- Believe it or not, even in economics, the concept of plane
**intersections**is used. For instance, in graphical methods, the**intersection**of supply and demand curves (which can be thought of as**planes**in multi-variable scenarios) determines equilibrium price and quantity.

- Believe it or not, even in economics, the concept of plane

*All images were created with GeoGebra.*