Equation of a Plane

Mathematically, a plane can be defined as a flat surface that is indefinite or vague. A plane is a two-dimensional surface made up of unlimited points and lines. They have an infinite as well as independent existence. Likewise, the lines and other geometric shapes, a plane also has different equations. The equations of a plane are mainly derived for a three-dimensional surface. These equations are used to represent and verify the shape of the respective dimensional surface. In mathematics, the study of a plane is necessary for all aspects. It is used in trigonometry, graphs, geometry, etc. This article explains everything about a plane and its equations.

Equations of a Plane

A plane is more difficult to describe as compared to lines and points. A completely perpendicular vector to the respective plane can help us derive its direction. All these factors togetherly form an equation to derive the plane’s direction.

Vector Equations of a Plane

Let us consider a plane with points P0(x₀, y₀, z₀) and a vector n orthogonal to the plane itself. This vector is called a normal vector. Assume the point P(x, y, z) as an arbitrary point on the plane and r0 and r as the position vectors of the points P0 and P. Hence, the vector r – r₀ can be represented in the plane. A vector equation is formed because the normal vector is orthogonal to r – r₀. It is written as, n .(r – r₀) = 0 or n . r = n . r₀

Scalar Equations of a Plane

A scalar equation can be formed after reconsidering a few values. To do so, we need to consider n = (a, b, c), r = (x, y, z), and r₀as (x₀, y₀, z₀). Hence, by substituting these values in the vector equation, we get, (a, b, c). (x – x₀, y – y₀, z – z₀) =0 It can be further simplified as, a (x – x₀) + b (y – y₀) + c (z – z₀) = 0 This equation, hence formed, is known as a scalar equation.

Linear Equation

The scalar equation can be rewritten in a different form. For that, we need to consider a new term, i.e., d = – (ax0 + by0 + cz0). The resulting equation is called a linear equation. It can be written as, ax + by + cz + d = 0 (Note: If a, b, c are not equal to 0, then we can consider that the linear equation hence formed represents a plane that has a normal vector.)

Important Points

• If the normal vectors of two planes are parallel, then the planes themselves are also parallel to each other.
• If two planes intersect in a straight line, an acute angle is formed between the two normal vectors.

Examples

1. Determine the equation of a plane through the point (2, 4, -1) with a normal vector n = (2, 3, 4).

Solution: By using the scalar equation, a (x – x₀) + b (y – y₀) + c (z – z₀) = 0 a = 2, b = 3, c = 4, x₀ = 2, y0 = 4, z₀ = -1 Therefore, by substituting these values in the scalar equation, we get, 2 (x – 2) + 3 (y – 4) + 4 (z – (-1)) = 0 By simplifying, 2x – 4 + 3y – 12 + 4z + 4 = 0 2x + 3y + 4z = 12 + 4 – 4 2x + 3y + 4z = 12 Therefore, 2x + 3y + 4z = 12 is the equation of the respective plane.

1. A plane ax + by + cz + d = 0 is given with point P1 (x₁, y₁, z₁). You have to find an equation for the distance D from point P1 to the plane.

Solution: Consider the point P0 (x₀, y₀, z₀) as any point on the given plane and b as a vector that corresponds to these points P1 and P0. Therefore, b = (x₁ – x₀, y₁ – y₀, z₁ – z₀) Let us assume the distance from the point P1 is exactly equal to the scalar projection of b on the normal vector n. Hence, the equation of the distance between the point P1 and the plane is formed as, D = n.bn = a (x₁ – x₀) + b (y₁ – y₀) + c ( z₁ – z₀) a₂ + b₂ + c₂ = | (ax₁ + by₁ + cz₁) – (ax₀ + by₀ + cz₀) |a₂ + b₂ + c₂ Here, ax₀ + by₀ + c z₀ + d = 0 Therefore, the equation for the distance will be written as, D = | (ax₁ + by₁ + cz₁) – (ax₀ + by₀ + cz₀) |a₂ + b₂ + c₂

Conclusion

A plane is a space that is indefinite and vague. It is a collection of several points and lines. The direction and shape of a plane can be derived using different equations. These equations of a plane can be written in various forms such as vector equation, linear equation, and scalar equation. With the fundamental equations of a plane, we can also find out the distance between the points and lines in the given plane. The equations of a plane are essential in mathematics as they help us verify and represent the shape of the respective plane. Hence, it has many applications in geometry as well as trigonometry.