how to find a vector orthogonal to a plane

Daniel Parker

3 min read

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18-01-2025

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Finding a vector orthogonal (perpendicular) to a plane is a fundamental concept in linear algebra with applications in computer graphics, physics, and machine learning. This article will guide you through several methods to achieve this, from using the normal vector to employing cross products. We'll explain the concepts clearly and provide examples to solidify your understanding.

Understanding Planes and Normals

Before diving into the methods, let's establish some foundational knowledge. A plane in three-dimensional space can be defined by a point on the plane and a vector orthogonal to it, called the normal vector. The normal vector points directly "out" from the plane. Think of it like the direction a plane's surface is facing.

Defining a Plane

A plane can be defined in several ways:

• Point and Normal Vector: Given a point P₀(x₀, y₀, z₀) on the plane and a normal vector n = <a, b, c>, the equation of the plane is: a(x - x₀) + b(y - y₀) + c(z - z₀) = 0.

• Three Non-Collinear Points: If you have three points that aren't all on the same line, you can find two vectors lying in the plane and then take their cross product to get the normal vector (as described below).

Methods for Finding an Orthogonal Vector

Here are the primary methods for finding a vector orthogonal to a plane:

Method 1: Using the Normal Vector Directly

The easiest method is when the plane's equation is already given in the form ax + by + cz = d. In this case, the normal vector is simply n = <a, b, c>. This vector is, by definition, orthogonal to the plane.

Example:

Let's say the equation of the plane is 2x + 3y - z = 5. The normal vector is n = <2, 3, -1>. Any scalar multiple of this vector (e.g., <4, 6, -2>) will also be orthogonal to the plane.

Method 2: Using the Cross Product (for three points)

If you're given three points on the plane, P₁, P₂, and P₃, you can find two vectors lying within the plane:

1. Create vectors: Calculate vectors v₁ = P₂ - P₁ and v₂ = P₃ - P₁. These vectors are within the plane.

2. Cross product: Compute the cross product of v₁ and v₂: n = v₁ x v₂. This resulting vector is orthogonal to both v₁ and v₂, and therefore orthogonal to the plane.

Example:

Let's say the points are P₁=(1, 0, 0), P₂=(0, 1, 0), and P₃=(0, 0, 1).

1. v₁ = P₂ - P₁ = <-1, 1, 0>
2. v₂ = P₃ - P₁ = <-1, 0, 1>
3. n = v₁ x v₂ = <1, 1, 1> This vector <1, 1, 1> is orthogonal to the plane defined by those three points.

Method 3: From Two Vectors in the Plane

If you already have two linearly independent vectors, u and v, that lie in the plane, their cross product will give you a vector orthogonal to the plane. This is essentially the same as Method 2 but presented differently. The cross product is defined as:

n = u x v = <(u₂v₃ - u₃v₂), (u₃v₁ - u₁v₃), (u₁v₂ - u₂v₁)>

Choosing the Right Method

The best method depends on how the plane is defined:

• Use Method 1 if the equation of the plane is readily available in the form ax + by + cz = d.
• Use Method 2 or 3 if you have three points or two vectors defining the plane.

Applications

Finding orthogonal vectors to planes has numerous practical applications:

• Computer Graphics: Calculating surface normals for lighting and shading.
• Physics: Determining the direction of a force acting on a surface.
• Machine Learning: Used in dimensionality reduction techniques and support vector machines.

By mastering these methods, you gain a powerful tool for tackling various problems involving planes and vectors in three-dimensional space. Remember to always check your results to ensure the calculated vector is indeed perpendicular to the plane.