Equation of a Plane Passing Through the Intersection of Two Given Planes
Consider two planes with normal vectors ( [Tex]\vec{n}_1 [/Tex]= ❬a1, b1, c1❭) and ([Tex] \vec{n}_2 [/Tex]= ❬a2, b2, c2❭ ), and distances from the origin ( d1 ) and ( d2 ) respectively.
The equation of the first plane is given by ( [Tex]\vec{r} \cdot \vec{n}_1 = d_1 [/Tex]), and the equation of the second plane is ( [Tex]\vec{r} \cdot \vec{n}_2 [/Tex]= d2 ).
Now, find the equation of the plane passing through their intersection. Since the plane is passing through the intersection of both planes, any point ( [Tex]\vec{r} [/Tex]) lying on it must satisfy both equations simultaneously.
Therefore, we can take the sum of the equations of the two planes and introduce a parameter (ƛ) to account for all possible points of intersection:
[Tex]\vec{r} \cdot (\vec{n}_1 + \lambda \vec{n}_2) = d_1 + \lambda d_2 [/Tex]
On expanding this equation, we get:
(x,y,z) · (a1 + ƛa2, b1 + ƛb2, c1 + ƛc2 = d1 + ƛd2
⇒ (x(a1 + ƛa2) +y(b1 + ƛb2) +z(c1 +ƛc2)) = d1 + ƛd2
⇒ a1x +a2 ƛx + b1y + b2 ƛy + c1z + c2 ƛz = d1 + ƛ d2
⇒ (a1x + b1y + c1z) + ƛ(a2x + b2y + c2z) = d1 + ƛ d2
This equation represents the plane passing through the intersection of the two given planes. The coefficients (a1, b1, c1) represent the normal vector of the first plane, (a2, b2, c2) represent the normal vector of the second plane, and ( ƛ) is a parameter representing the different points of intersection along the line of intersection of 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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