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
- 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.
- 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.