Intercept Form of Equation of Plane
To derive the equation of a plane in intercept form, we start with the general form of the equation of a plane:
ax + by + cz + d = 0
where ( a ), ( b ), and ( c ) are the coefficients representing the direction of the plane’s normal vector, and ( d ) is the distance from the origin along the normal vector.
To obtain the intercept form, we divide the entire equation by ( -d ) to isolate ( x ), ( y ), and ( z ):
[Tex]\frac{x}{-\frac{d}{a}} + \frac{y}{-\frac{d}{b}} + \frac{z}{-\frac{d}{c}} = 1 [/Tex]
Simplifying each term, we get:
[Tex]\frac{x}{\frac{-d}{a}} + \frac{y}{\frac{-d}{b}} + \frac{z}{\frac{-d}{c}} = 1[/Tex]
⇒ [Tex]\frac{x}{\frac{a}{-d}} + \frac{y}{\frac{b}{-d}} + \frac{z}{\frac{c}{-d}} = 1[/Tex]
⇒ [Tex]\frac{x}{\frac{a}{d}} + \frac{y}{\frac{b}{d}} + \frac{z}{\frac{c}{d}} = 1 [/Tex]
Finally, we rewrite the equation in the intercept form:
[Tex]\frac{x}{\frac{d}{a}} + \frac{y}{\frac{d}{b}} + \frac{z}{\frac{d}{c}} = 1 [/Tex]
Hence, the equation of a plane in intercept form is:
[Tex]\frac{x}{\frac{d}{a}} + \frac{y}{\frac{d}{b}} + \frac{z}{\frac{d}{c}} = 1 [/Tex]
Example of Planes in Intercept Form
In intercept form, the equation of a plane is given by its intercepts on the three coordinate axes. Here are some examples:
XY-Plane: ax + by + cz = 1
- This equation represents a plane intersecting the x-axis at a, the y-axis at b, and the z-axis at c.
YZ-Plane: ax=0
- This equation represents a plane parallel to the yz-plane and intersecting the x-axis at a.
XZ-Plane: by=0
- This equation represents a plane parallel to the xz-plane and intersecting the y-axis at b.
Plane with Intercepts: x/2 + y/3 + z/4 = 1
- This equation represents a plane intersecting the x-axis at 2, the y-axis at 3, and the z-axis at 4.
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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