Equation of a Plane in Normal Form
Consider a plane in three-dimensional space, defined by a point (P(x1, y1, z1)) and a normal vector ([Tex]\vec{n} = \langle a, b, c \rangle[/Tex]).
The equation of a plane in normal form is given by ([Tex]\vec{r} \cdot \vec{n}[/Tex] = d), where ([Tex]\vec{r} = \langle x, y, z \rangle[/Tex]) represents any point on the plane, and (d) is the perpendicular distance from the origin to the plane.
Now, let ([Tex]\vec{r_0} = \langle x_1, y_1, z_1 \rangle[/Tex]) be a point on the plane.
The vector connecting a general point ([Tex]\vec{r}[/Tex]) on the plane to the given point ([Tex]\vec{r_0}[/Tex]) is ([Tex]\vec{r} – \vec{r_0}[/Tex]).
For a point ([Tex]\vec{r}[/Tex]) to lie on the plane, the normal vector ([Tex]\vec{n}[/Tex]) must be perpendicular to the vector ([Tex]\vec{r} – \vec{r_0})[/Tex]. Therefore, the dot product of these vectors should be zero.
Mathematically, ([Tex]\vec{r} – \vec{r_0}) \cdot \vec{n}[/Tex] = 0).
Expanding this, we get ([Tex]\langle x, y, z \rangle – \langle x_1, y_1, z_1 \rangle) \cdot \langle a, b, c \rangle[/Tex] = 0).
Simplifying further, we have (a(x – x1) + b(y – y1) + c(z – z1) = 0).
Now, rearrange the terms to obtain the normal form of the plane equation: (ax + by + cz = ax1 + by1 + cz1).
Thus, 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,
- \vec{r} = ❬ x, y, z ❭ represents a generic point in the plane,
- \vec{r_0} = ❬ x0, y0, z0 ❭ is a specific point in the plane, and
- \vec{n} = ❬ a, b, c ❭ is the normal vector to the plane.
This equation represents the plane in normal form, where (a, b, c) is the normal vector to the plane, and (d = ax1 + by1 + cz1) is the perpendicular distance from the origin to the plane.
Examples of Planes in Point-Normal Form
Here are examples of planes in point-normal form:
Plane Parallel to the XY Plane
- Equation: z – 3 = 0
- Normal Vector: ❬ 0, 0, 1 ❭
- Point in the Plane: ❬ x, y, 3 ❭
Inclined Plane in First Quadrant
- Equation: 2x – y + z – 5 = 0
- Normal Vector: ❬ 2, -1, 1 ❭
- Point in the Plane: ❬ x, 2x + 3, 3x – 2❭
Vertical Plane Passing Through the Y-Axis
- Equation: x + z + 4 = 0
- Normal Vector: ❬ 1, 0, 1 ❭
- Point in the Plane: ❬ -4, y, -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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