A plane, along with a point and a straight line, is a basic geometric element. With its use, many figures are constructed in spatial geometry. In this article, we will consider in more detail the question of how to find the angle between two planes.
The concept
Before talking about the angle between two planes, you should well understand which element in the geometry we are talking about. We will understand the terminology. A plane is an endless collection of points in space, connecting together which we get vectors. The latter will be perpendicular with respect to a single vector. It is commonly called the normal to the plane.
The figure above shows the plane and two normal vectors to it. It can be seen that both vectors lie on one straight line. The angle between them is 180 o .
Equations
The angle between two planes can be determined if the mathematical equation of the geometric element in question is known. There are several types of similar equations, the names of which are listed below:
- general type;
- vectorial;
- in segments.
These three types are the most convenient for solving various kinds of tasks, so they are most often used.
A general equation is as follows:
A * x + B * y + C * z + D = 0.
Here x, y, z are the coordinates of an arbitrary point belonging to a given plane. Parameters A, B, C, and D are numbers. The convenience of this form of writing is that the numbers A, B, C are the coordinates of the vector normal to the plane.
The vector shape of the plane can be represented as follows:
x, y, z) = (x 0 , y 0 , z 0 ) + α * (a 1 , b 1 , c 1 ) + β * (a 2 , b 2 , c 2 ).
Here (a 2 , b 2 , c 2 ) and (a 1 , b 1 , c 1 ) are the parameters of two coordinate vectors that belong to the plane under consideration. The point (x 0 , y 0 , z 0 ) also lies in this plane. Parameters α and β can take independent from each other and arbitrary values.
Finally, the plane equation in segments is represented in the following mathematical form:
x / p + y / q + z / l = 1.
Here p, q, l are specific numbers (including negative ones). This type of equation is convenient when you need to depict a plane in a rectangular coordinate system, since the numbers p, q, l show the points of intersection with the x, y, and z axes of the plane.
Note that each type of equation can be transformed into any other using simple mathematical operations.
The formula for the angle between two planes
Now consider the following nuance. In three-dimensional space, two planes can be located in just two ways. Either intersect or be parallel. Between two planes, an angle is called what is located between their vectors guides (normal). Intersecting, 2 vectors form 2 angles (sharp and dull in the general case). The angle between the planes is considered to be sharp. Consider the equation.
The formula for the angle between two planes is:
θ = arccos (| (n 1 ¯ * n 2 ¯) | / (| n 1 ¯ | * | n 2 ¯ |)).
It is easy to guess that this expression is a direct consequence of the scalar product of normal vectors n 1 ¯ and n 2 ¯ for the planes in question. The modulus of the scalar product in the numerator indicates that the angle θ will take only values from 0 o to 90 o . The product of the moduli of normal vectors in the denominator means the product of their lengths.
Note that if (n 1 ¯ * n 2 ¯) = 0, then the planes intersect at right angles.
Task example
Having figured out what is called the angle between two planes, we solve the following problem. As an example. So, it is necessary to calculate the angle between such planes:
2 * x - 3 * y + 4 = 0;
(x, y, z) = (2, 0, -1) + α * (1, 1, -1) + β * (0, 2, 3).
To solve the problem, it is necessary to know the directing vectors of the planes. For the first plane, the normal vector is: n 1 ¯ = (2, -3, 0). To find the normal vector of the second plane, multiply the vectors after the parameters α and β. As a result, we obtain the vector: n 2 ¯ = (5, -3, 2).
To determine the angle θ, we use the formula from the previous paragraph. We get:
θ = arccos (| ((2, -3, 0) * (5, -3, 2)) | / (| (2, -3, 0) | * | (5, -3, 2) |)) =
= arccos (19 / √ (13 * 38)) = 0.5455 rad.
The calculated angle in radians corresponds to 31.26 o . Thus, the plane of the conditions of the problem intersect at an angle of 31.26 o .