Equation of a Plane Passing Through Three Points
To derive the equation of a plane passing through three non-collinear points ( [Tex]\vec{a} = [/Tex] ❬x1,y1,z1❭), ( [Tex]\vec{b} =[/Tex] ❬x2, y2, z2❭), and ( [Tex]\vec{c} = [/Tex]❬x3,y3,z3❭), we will use the cross product.
First, define two vectors that lie in the plane, formed by the given points ( [Tex]\vec{a}[/Tex] ), ( [Tex]\vec{b} [/Tex]), and ( [Tex]\vec{c}[/Tex] ). We’ll call these vectors ( [Tex]\vec{v_1}[/Tex] ) and ( [Tex]\vec{v_2}[/Tex]):
[Tex]\vec{v_1} = \vec{b} – \vec{a} = [/Tex]❬ x2 – x1, y2 – y1, z2 – z1 ❭
[Tex]\vec{v_2} = \vec{c} – \vec{a} = [/Tex]❬x3 – x1, y3 – y1, z3 – z1❭
Next, we take the cross product of ( [Tex]\vec{v_1}[/Tex] ) and ( [Tex]\vec{v_2}[/Tex] ) to find a vector that is perpendicular to the plane:
[Tex]\vec{n} = \vec{v_1} \times \vec{v_2} [/Tex]
[Tex]\Rightarrow \vec{n} =\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x_2 – x_1 & y_2 – y_1 & z_2 – z_1 \\ x_3 – x_1 & y_3 – y_1 & z_3 – z_1 \end{vmatrix} [/Tex]
[Tex]\Rightarrow \vec{n} [/Tex] = (y2 – y1)(z3 – z1) – (z2 – z1)(y3 – y1) [Tex]\hat{i}[/Tex] – (x2 – x1)(z3 – z1) + (z2 – z1)(x3 – x1) [Tex]\hat{j}[/Tex] + (x2 – x1)(y3 – y1) – (y2 – y1)(x3 – x1) [Tex]\hat{k} [/Tex]
Now, the equation of the plane passing through the points ( [Tex]\vec{a}[/Tex] ), ( [Tex]\vec{b}[/Tex] ), and ( [Tex]\vec{c}[/Tex] ) is given by:
[Tex]\vec{n} \cdot (\vec{r} – \vec{a})[/Tex] = 0
Substituting the components of ( [Tex]\vec{n}[/Tex] ) and ( [Tex]\vec{a} [/Tex]) into the equation:
(y2 – y1)(z3 – z1) – (z2 – z1)(y3 – y1)(x – x1) + (z2 – z1)(x3 – x1)(y – y1) + (x2 – x1)(y3 – y1) – (y2 – y1)(x3 – x1)(z – z1) = 0
This equation represents the plane passing through the three non-collinear points [Tex]\vec{a}[/Tex], [Tex]\vec{b}[/Tex], and [Tex]\vec{c} [/Tex].
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
Contact Us