General Form of Equation of a Plane
The general form of the equation of a plane is ( ax + by + cz + d = 0 ), where ( a ), ( b ), and ( c ) are constants representing the plane’s normal vector, and ( d ) is a constant representing the plane’s distance from the origin.
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:
- Cartesian Form: Ax + By + Cz = D Where A, B, and C are the coefficients of x, y, and z respectively, and D is a constant.
- Parametric Form:[Tex]\vec{r}[/Tex] = [Tex]\vec{a}[/Tex] + s[Tex]\vec{u}[/Tex] + t[Tex]\vec{v}[/Tex]. Where r is a position vector, a is a point on the plane, and u and v are direction vectors parallel to the plane. s and t are scalar parameters.
- Normal Form: Ax + By + Cz + D = 0, Where A, B, and C are the coefficients of x, y, and z respectively, and (A,B,C) is the normal vector to the plane. D is the distance from the origin to the plane along the direction of the normal vector.
Examples of Planes in General Form
Some examples of planes represented in general form Ax + By + Cz + D = 0:
Horizontal Plane: z=0 (or equivalently, 0x + 0y + z + 0 = 0
- This equation represents the xy-plane, where all points have z-coordinate equal to zero.
Vertical Plane: x=2 (or equivalently, x + 0y + 0z − 2 = 0
- This equation represents a vertical plane parallel to the yz-plane, passing through the point (2, 0, 0).
Diagonal Plane: 2x − 3y + z − 5 = 0
- This equation represents a plane with coefficients A = 2, B = −3, C = 1, and D = −5, passing through the origin and inclined with respect to the coordinate axes.
Perpendicular Plane: 3x + 4y − 2z + 6 = 0
- This equation represents a plane perpendicular to the vector n=(3, 4, −2), passing through the point (−2, 3, 0).
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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