Equation of a Plane in Three Dimensional Space
The equation of a plane in three-dimensional space can be expressed in various forms, each serving different purposes. Here are the main forms of the equation of a plane
General Form: The general form of the equation of a plane is represented as:
Ax + By + Cz + D = 0
Where ( A ), ( B ), ( C ), and ( D ) are constants, and ( x ), ( y ), and ( z ) are the variables representing coordinates in three-dimensional space.
Point-Normal Form: The point-normal form of the equation of a plane is given by:
[Tex]\vec{n} \cdot (\vec{r} – \vec{r_0}) = 0[/Tex]
Where ( [Tex]\vec{n}[/Tex] ) is the normal vector to the plane, ( [Tex]\vec{r_0}[/Tex] ) is a known point on the plane, and ( [Tex]\vec{r}[/Tex] ) represents any point on the plane.
Intercept Form: The intercept form of the equation of a plane is expressed as:
[Tex]\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1[/Tex]
Where ( a ), ( b ), and ( c ) are the intercepts of the plane on the ( x ), ( y ), and ( z ) axes respectively.
Vector Form: The vector form of the equation of a plane is represented as:
[Tex]\vec{r} = \vec{r_0} + s\vec{v} + t\vec{w}[/Tex]
Where ( [Tex]\vec{r_0}[/Tex] ) is a known point on the plane, and ( [Tex]\vec{v}[/Tex] ) and ( [Tex]\vec{w}[/Tex] ) are two non-parallel vectors lying on the plane, and s and t are scalar parameters.
Equation for Intersection of Planes
The equation for the intersection of two planes can be found by solving their respective equations simultaneously.
Consider two planes represented by the equations (ax + by + cz + d1 = 0) and (ex + fy + gz + d2 = 0). To find the intersection line or point, solve these equations together to determine values for (x), (y), and (z) that satisfy both planes.
For example, if we have planes with equations (2x + 3y – z + 5 = 0) and (4x – y + 2z – 8 = 0), solving them simultaneously gives the values (x = 2), (y = 1), and (z = 3). These values represent a point of intersection for the two planes.
Equation of Plane
Equation of Plane describes its position and orientation in three-dimensional space, typically represented in the form (ax + by + cz + d = 0), where (a), (b), and (c) are coefficients representing the plane’s normal vector, and (d) is the distance from the origin along the normal vector.
In this article, we will learn about the what is the equation of a plane, its definition and general form the equation, the equation of a plane in 3D Space, a Cartesian form of an equation of a plane, the equation of a plane in intercept and parametric form, etc. At the end of this article, you will see some examples of solved problems that will provide a better understanding of the topic.
Table of Content
- What is the Equation of Plane?
- General Form of Equation of a Plane
- Equation of a Plane in Three Dimensional Space
- Methods to Find Equation of a Plane
- Equation of a Plane in Normal Form
- Equation of a Plane Passing Through Three Points
- Cartesian Form of Equation of a Plane
- Equation of a Plane in Parametric Form
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