Calculate the angle between a line and a plane. Coordinate method for solving problems

2024 Author: Angel Austin | [email protected]. Last modified: 2024-01-02 17:39

One of the most common problems in stereometry is the intersection of straight lines and planes and the calculation of the angles between them. Let us consider in this article in more detail the so-called coordinate method and the angles between the line and the plane.

Line and plane in geometry

Before considering the coordinate method and the angle between a line and a plane, you should get acquainted with the named geometric objects.

A line is such a collection of points in space or on a plane, each of which can be obtained by linearly transferring the previous one to a certain vector. In what follows, we denote this vector by the symbol u¯. If this vector is multiplied by any number that is not equal to zero, then we get a vector parallel to u¯. A line is a linear infinite object.

A plane is also a set of points located in such a way that if they are made up of arbitrary vectors, then all of them will be perpendicular to some vector n¯. The latter is called normal or simply normal. A plane, unlike a straight line, is a two-dimensional infinite object.

Coordinate method for solving geometry problems

Based on the name of the method itself, we can conclude that we are talking about a method of solving problems, which is based on the performance of analytical sequential calculations. In other words, the coordinate method allows you to solve geometric problems using universal algebra tools, the main of which are equations.

It should be noted that the method under consideration appeared at the dawn of modern geometry and algebra. A great contribution to its development was made by Rene Descartes, Pierre de Fermat, Isaac Newton and Leibniz in the 17th-18th centuries.

The essence of the method is to calculate the distances, angles, areas and volumes of geometric elements based on the coordinates of known points. Note that the form of the final equations obtained depends on the coordinate system. Most often, the rectangular Cartesian system is used in problems, since it is most convenient to work with.

Line Equation

Consideration of the coordinate method and the angles between the line and the plane, let's start with setting the equation of the line. There are several ways to represent lines in algebraic form. Here we consider only the vector equation, since it can be easily obtained from it in any other form and is easy to work with.

Assume that there are two points: P and Q. It is known that a line can be drawn through them, and itwill be the only one. The corresponding mathematical representation of the element looks like this:

(x, y, z)=P + λPQ¯.

Where PQ¯ is a vector whose coordinates are obtained as follows:

PQ¯=Q - P.

The symbol λ denotes a parameter that can take absolutely any number.

In the written expression, you can change the direction of the vector, and also substitute the coordinates Q instead of the point P. All these transformations will not lead to a change in the geometric location of the line.

Note that when solving problems, it is sometimes required to represent the written vector equation in an explicit (parametric) form.

Setting a plane in space

As well as for a straight line, there are also several forms of mathematical equations for a plane. Among them, we note the vector, the equation in segments and the general form. In this article, we will pay special attention to the last form.

A general equation for an arbitrary plane can be written as follows:

Ax + By + Cz + D=0.

Latin capital letters are certain numbers that define a plane.

The convenience of this notation is that it explicitly contains a vector normal to the plane. It is equal to:

n¯=(A, B, C).

Knowing this vector makes it possible, by looking briefly at the equation of the plane, to imagine the location of the latter in the coordinate system.

Mutual arrangement inspace of line and plane

In the next paragraph of the article we will move on to the consideration of the coordinate method and the angle between the line and the plane. Here we will answer the question of how the considered geometric elements can be located in space. There are three ways:

The straight line intersects the plane. Using the coordinate method, you can calculate at what single point the line and the plane intersect.
The plane of a straight line is parallel. In this case, the system of equations of geometric elements has no solution. To prove parallelism, the property of the scalar product of the directing vector of the straight line and the normal of the plane is usually used.
The plane contains a line. Solving the system of equations in this case, we will come to the conclusion that for any value of the parameter λ, the correct equality is obtained.

In the second and third cases, the angle between the specified geometric objects is equal to zero. In the first case, it lies between 0 and 90o.

Calculation of angles between lines and planes

Now let's go directly to the topic of the article. Any intersection of a line and a plane occurs at some angle. This angle is formed by the straight line itself and its projection onto the plane. A projection can be obtained if from any point of a straight line a perpendicular is lowered onto the plane, and then through the obtained point of intersection of the plane and the perpendicular and the point of intersection of the plane and the original line, draw a straight line that will be a projection.

Calculating the angles between lines and planes is not a difficult task. To solve it, it is enough to know the equations of the corresponding geometric objects. Let's say these equations look like this:

(x, y, z)=(x₀, y₀, z_{0) + λ(a, b, c);}

Ax + By + Cz + D=0.

The desired angle is easily found using the property of the product of the scalar vectors u¯ and n¯. The final formula looks like this:

θ=arcsin(|(u¯n¯)|/(|u¯||n¯|)).

This formula says that the sine of the angle between a line and a plane is equal to the ratio of the modulus of the scalar product of the marked vectors to the product of their lengths. To understand why sine appeared instead of cosine, let's turn to the figure below.

It can be seen that if we apply the cosine function, we will get the angle between the vectors u¯ and n¯. The desired angle θ (α in the figure) is obtained as follows:

θ=90o- β.

The sine appears as a result of applying the reduction formulas.

Example problem

Let's move on to the practical use of the acquired knowledge. Let's solve a typical problem on the angle between a straight line and a plane. The following coordinates of four points are given:

P=(1, -1, 0);

Q=(-1, 2, 2);

M=(0, 3, -1);

N=(-2, -1, 1).

It is known that through points PQMa plane passes through it, and a straight line passes through MN. Using the coordinate method, the angle between the plane and the line must be calculated.

First, let's write down the equations of the straight line and the plane. For a straight line, it is easy to compose it:

MN¯=(-2, -4, 2)=>

(x, y, z)=(0, 3, -1) + λ(-2, -4, 2).

To make the equation of the plane, we first find the normal to it. Its coordinates are equal to the vector product of two vectors lying in the given plane. We have: