How to Find a Plane Perpendicular to Another Plane

How to Find a Plane Perpendicular to Another Plane

In geometry, a plane is a flat surface that extends infinitely in all directions. Two planes are perpendicular to each other if they intersect at a right angle. Finding the equation of a plane perpendicular to another plane can be a useful tool in a variety of applications, such as engineering, architecture, and manufacturing.

In this article, we will discuss the steps involved in finding a plane perpendicular to another plane. We will also provide some examples to help you understand the process. So if you’re ever tasked with finding the equation of a plane perpendicular to another plane, you’ll know exactly what to do.

Step 1: Find the normal vector to the first plane.

The first step is to find the normal vector to the first plane. The normal vector is a vector that is perpendicular to the plane. It can be found by taking the cross product of any two vectors that lie in the plane.

Step 2: Find the equation of the second plane.

Once you have the normal vector to the first plane, you can use it to find the equation of the second plane. The equation of a plane is given by “`ax + by + cz + d = 0“`, where “`a“`, “`b“`, and “`c“` are the coefficients of the normal vector, and “`d“` is the constant term.

Step 3: Check that the two planes are perpendicular.

To check that the two planes are perpendicular, you can take the dot product of their normal vectors. If the dot product is equal to zero, then the two planes are perpendicular.

Example 1: Find the equation of the plane that is perpendicular to the plane “`2x + 3y + 4z = 5“` and passes through the point “`(1, 2, 3)“`.

Solution:

1. Find the normal vector to the first plane.

“`
n = <2, 3, 4>
“`

2. Find the equation of the second plane.

“`
2x + 3y + 4z + d = 0
“`

3. Substitute the point “`(1, 2, 3)“` into the equation of the second plane and solve for “`d“`.

“`
2(1) + 3(2) + 4(3) + d = 0
“`

“`
14 + d = 0
“`

“`
d = -14
“`

4. Substitute the value of “`d“` into the equation of the second plane.

“`
2x + 3y + 4z – 14 = 0
“`

This is the equation of the plane that is perpendicular to the plane “`2x + 3y + 4z = 5“` and passes through the point “`(1, 2, 3)“`.

Step	Description	Example
1	Find the normal vector of the first plane.	The normal vector of a plane is a vector that is perpendicular to the plane. To find the normal vector of a plane, you can use the following formula: \[ n = \begin{bmatrix} A \\ B \\ C \end{bmatrix} \] where A, B, and C are the coefficients of the plane’s equation in standard form: \[ Ax + By + Cz = D \]
2	Find the cross product of the normal vectors of the two planes.	The cross product of two vectors is a vector that is perpendicular to both of the original vectors. To find the cross product of two vectors, you can use the following formula: \[ v \times w = \begin{vmatrix} i & j & k \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} \] where u and v are the two vectors.
3	The resulting vector is the normal vector of the plane that is perpendicular to both of the original planes.	In the example above, the normal vectors of the two planes are \[ n_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \] and \[ n_2 = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} \] The cross product of these two vectors is \[ n_1 \times n_2 = \begin{vmatrix} i & j & k \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix} = \begin{bmatrix} -10 \\ 20 \\ -10 \end{bmatrix} \] This vector is the normal vector of the plane that is perpendicular to both of the original planes.

Step

Description

Example

Find the normal vector of the first plane.

The normal vector of a plane is a vector that is perpendicular to the plane.

To find the normal vector of a plane, you can use the following formula:

\[
n = \begin{bmatrix}
A \\
B \\
C
\end{bmatrix}
\]

where A, B, and C are the coefficients of the plane’s equation in standard form:

\[
Ax + By + Cz = D
\]

Find the cross product of the normal vectors of the two planes.

The cross product of two vectors is a vector that is perpendicular to both of the original vectors.

To find the cross product of two vectors, you can use the following formula:

\[
v \times w =
\begin{vmatrix}
i & j & k \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}
\]

where u and v are the two vectors.

The resulting vector is the normal vector of the plane that is perpendicular to both of the original planes.

In the example above, the normal vectors of the two planes are

\[
n_1 = \begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix}
\]

and

\[
n_2 = \begin{bmatrix}
4 \\
5 \\
6
\end{bmatrix}
\]

The cross product of these two vectors is

\[
n_1 \times n_2 =
\begin{vmatrix}
i & j & k \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix}
=
\begin{bmatrix}
-10 \\
20 \\
-10
\end{bmatrix}
\]

This vector is the normal vector of the plane that is perpendicular to both of the original planes.

A plane perpendicular to another plane is a plane that intersects the other plane at a right angle. In other words, the angle between the two planes is 90 degrees. Planes that are perpendicular to each other are also called orthogonal planes.

Planes can be perpendicular to each other in two ways:

Directly perpendicular: Two planes are directly perpendicular if they intersect at a right angle and do not share any points.
Obliquely perpendicular: Two planes are obliquely perpendicular if they intersect at a right angle but do share some points.

In this tutorial, we will focus on finding planes that are directly perpendicular to each other.

What is a Plane Perpendicular to Another Plane?

A plane perpendicular to another plane is a plane that intersects the other plane at a right angle. In other words, the angle between the two planes is 90 degrees.

Planes that are perpendicular to each other are also called orthogonal planes.

How to Find a Plane Perpendicular to Another Plane?

There are several ways to find a plane perpendicular to another plane. One way is to use the following steps:

1. Find the normal vector to the first plane. The normal vector to a plane is a vector that is perpendicular to the plane. To find the normal vector to a plane, you can use the following formula:

“`
n =
“`

where `a`, `b`, and `c` are the coefficients of the plane equation in the form `ax + by + cz = d`.

2. Find the intersection of the normal vector with the second plane. The intersection of the normal vector with the second plane will be a point that lies on the plane that is perpendicular to the first plane.

3. Find the equation of the plane that passes through the point of intersection and is perpendicular to the first plane. The equation of the plane can be found using the following formula:

“`
ax + by + cz = d
“`

where `a`, `b`, and `c` are the coefficients of the plane equation and `d` is the constant term.

Example

Let’s find a plane that is perpendicular to the plane `2x + 3y + 4z = 6`.

1. Find the normal vector to the plane. The normal vector to the plane is `<2, 3, 4>`.

2. Find the intersection of the normal vector with the second plane. The intersection of the normal vector with the plane `x + y + z = 1` is the point `(1, 1, 1)`.

3. Find the equation of the plane that passes through the point of intersection and is perpendicular to the first plane. The equation of the plane is `2x + 3y + 4z = 6`.

In this tutorial, we learned how to find a plane that is perpendicular to another plane. We used the following steps:

1. Find the normal vector to the first plane.
2. Find the intersection of the normal vector with the second plane.
3. Find the equation of the plane that passes through the point of intersection and is perpendicular to the first plane.

How To Find A Plane Perpendicular To Another Plane?

In this tutorial, we will show you how to find a plane that is perpendicular to another plane. We will use the following steps:

1. Find the normal vector to the first plane.
2. Find the equation of the first plane.
3. Find two points that lie in the first plane.
4. Find the vector that is perpendicular to the first plane and passes through one of the points.
5. Find the equation of the plane that is perpendicular to the first plane and passes through the two points.

1. Find the normal vector to the first plane

The normal vector to a plane is a vector that is perpendicular to the plane. To find the normal vector to a plane, we can use the following formula:

$n=\\frac{\\vec{a}\\times\\vec{b}}{\\left|\\vec{a}\\times\\vec{b}\\right|}$

where $\\vec{a}$ and $\\vec{b}$ are two vectors that lie in the plane.

2. Find the equation of the first plane

The equation of a plane can be written in the following form:

$ax by cz d=0$

where $a,b,c$ are the coefficients of the plane and $d$ is the constant term.

To find the equation of the first plane, we can use the following steps:

1. Choose two points that lie in the plane.
2. Find the vector that is perpendicular to the plane and passes through one of the points.
3. Use the vector to find the coefficients of the plane.

3. Find two points that lie in the first plane

To find two points that lie in the first plane, we can use the following steps:

1. Choose any point in the plane.
2. Move a small distance in the direction of the normal vector to the plane.
3. Choose the point that you land on.

4. Find the vector that is perpendicular to the first plane and passes through one of the points

To find the vector that is perpendicular to the first plane and passes through one of the points, we can use the following steps:

1. Choose a point in the plane.
2. Find the normal vector to the plane.
3. Use the cross product to find the vector that is perpendicular to the plane and passes through the point.

5. Find the equation of the plane that is perpendicular to the first plane and passes through the two points

To find the equation of the plane that is perpendicular to the first plane and passes through the two points, we can use the following steps:

1. Find the vector that is perpendicular to the first plane and passes through one of the points.
2. Find the equation of the plane that passes through the two points and is perpendicular to the vector.

In this tutorial, we showed you how to find a plane that is perpendicular to another plane. We used the following steps:

How do I find a plane perpendicular to another plane?

There are a few ways to find a plane perpendicular to another plane. One way is to use the cross product. The cross product of two vectors is a vector that is perpendicular to both of them. So, if you have two vectors that lie in the same plane, you can find a vector that is perpendicular to both of them by taking their cross product.

Another way to find a plane perpendicular to another plane is to use the normal vector. The normal vector to a plane is a vector that is perpendicular to the plane. You can find the normal vector to a plane by taking the cross product of two vectors that lie in the plane.

Finally, you can also find a plane perpendicular to another plane by using the equation of the plane. The equation of a plane is given by Ax + By + Cz + D = 0, where A, B, C, and D are constants. If you have the equation of one plane, you can find the equation of a plane that is perpendicular to it by multiplying the equation of the first plane by a constant and then adding a constant to both sides.

What is the equation of a plane perpendicular to another plane?

The equation of a plane perpendicular to another plane is given by Ax + By + Cz + D = 0, where A, B, C, and D are constants. To find the equation of a plane perpendicular to another plane, you can multiply the equation of the first plane by a constant and then add a constant to both sides.

For example, if the equation of the first plane is 2x – 3y + 4z = 5, then the equation of a plane perpendicular to it is -4x + 6y – 8z = -10.

How do I find the angle between two planes?

The angle between two planes can be found using the following formula:

= arccos(|nn|/(|n| |n|))

where is the angle between the planes, n and n are the normal vectors to the planes, and |n| and |n| are the magnitudes of the normal vectors.

To find the normal vectors to the planes, you can use the cross product. The cross product of two vectors is a vector that is perpendicular to both of them. So, if you have two vectors that lie in the same plane, you can find a vector that is perpendicular to both of them by taking their cross product.

Once you have the normal vectors to the planes, you can find the angle between them using the formula above.

What are some applications of finding a plane perpendicular to another plane?

There are many applications of finding a plane perpendicular to another plane. Some of these applications include:

Computer graphics: In computer graphics, planes are often used to represent surfaces. Finding a plane perpendicular to another plane can be used to create shadows or reflections.

Engineering: In engineering, planes are often used to represent components of a design. Finding a plane perpendicular to another plane can be used to check for interference between components or to find the best way to join two components together.

Physics: In physics, planes are often used to represent fields. Finding a plane perpendicular to another plane can be used to find the direction of the field or to find the magnitude of the field.

we have discussed the steps on how to find a plane perpendicular to another plane. We first need to find the normal vector of the given plane. Then, we can find the equation of the perpendicular plane by taking the dot product of the normal vector and the vector that is perpendicular to the given plane. Finally, we can graph the two planes to visualize their relationship.

We hope that this article has been helpful. Thank you for reading!

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Dale Richard
Dale, in his mid-thirties, embodies the spirit of adventure and the love for the great outdoors. With a background in environmental science and a heart that beats for exploring the unexplored, Dale has hiked through the lush trails of the Appalachian Mountains, camped under the starlit skies of the Mojave Desert, and kayaked through the serene waters of the Great Lakes.

His adventures are not just about conquering new terrains but also about embracing the ethos of sustainable and responsible travel. Dale’s experiences, from navigating through dense forests to scaling remote peaks, bring a rich tapestry of stories, insights, and practical tips to our blog.

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