One of the most common in stereometry is the problem of intersecting straight lines and planes and calculating the angles between them. We 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 the line and the plane, you should get acquainted with the named geometric objects.
A straight line is a collection of points in space or on a plane, each of which can be obtained by linearly transferring the previous one to a specific vector. We will denote this vector by u¯. If this vector is multiplied by any number that is not equal to zero, then we get a parallel u¯ vector. A straight line is a linear infinite object.
A plane is also a collection of points that are arranged in such a way that, if arbitrary vectors are composed of them, then all of them will be perpendicular to some vector n¯. The latter is called normal or just normal. A plane, in contrast to a straight line, is a two-dimensional infinite object.
Coordinate method for solving problems in geometry
Based on the name of the method itself, we can conclude that we are talking about a method for solving problems, which is based on performing 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 the emergence of modern geometry and algebra. A great contribution to its development was made by Rene Descartes, Pierre Fermat, Isaac Newton and Leibniz in the XVII-XVIII 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 resulting final equations depends on the coordinate system. Most often, a rectangular Cartesian system is used in tasks, since it is most convenient to work with it.
Straight line equation
We will begin the consideration of the coordinate method and the angles between the line and the plane by setting the equation of the line. There are several ways to represent straight lines in algebraic form. Here we consider only the vector equation, since any other form can be easily obtained from it and it is easy to work with it.
Suppose that there are two points: P and Q. It is known that a straight line can be drawn through them, and it will be unique. The corresponding mathematical representation of the element looks like this:
(x, y, z) = P + λ * PQ¯.
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 recorded expression, you can change the direction of the vector, and also substitute the coordinates of Q instead of the point P. All these transformations will not lead to a change in the geometric arrangement of the line.
Note that in solving problems it is sometimes required to represent the written vector equation in an explicit (parametric) form.
Defining a plane in space
As for a straight line, there are also several forms of mathematical equations for a plane. Among them, we note the vector equation in segments and the general form. In this article, we will pay particular attention to the latter form.
A general equation for an arbitrary plane can be written as follows:
A * x + B * y + C * z + D = 0.
Latin capital letters are certain numbers that define a plane.
The convenience of this form of writing is that it explicitly contains a vector normal to the plane. It is equal to:
n¯ = (A, B, C).
Knowing this vector allows you to glance at the equation of the plane and imagine the location of the latter in the coordinate system.
Mutual arrangement in space of a straight line and a plane
In the next paragraph of the article, we will move on to consider the coordinate method and the angle between the line and the plane. Here we will answer the question of how in space the considered geometric elements can be located. There are three such methods:
- A straight line crosses a plane. Using the coordinate method, you can calculate at which single point the line and the plane intersect.
- The plane of the 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 line and the normal to the plane is usually used.
- The plane contains a straight line. Solving the system of equations in this case, we conclude that for any value of the parameter λ we obtain the correct equality.
In the second and third cases, the angle between the indicated geometric objects is zero. In the first case, it lies in the range from 0 to 90 o .
Calculation of angles between lines and planes
Now we turn directly to the topic of the article. Any intersection of a line and a plane occurs at a certain angle. This angle is formed by the most direct and its projection onto the plane. A projection can be obtained if you drop a perpendicular from any point on the line to the plane, and then draw a line through the resulting intersection point of the plane and perpendicular and the intersection point of the plane and the original line, which will be the projection.
Calculating 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. Suppose these equations are as follows:
(x, y, z) = (x 0 , y 0 , z 0 ) + λ * (a, b, c);
A * x + B * y + C * z + D = 0.
The desired angle is easily found if we use 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 the line and the 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 the sine appeared instead of the cosine, we turn to the figure below.
It can be seen that if we apply the cosine function, then we get the angle between the vectors u¯ and n¯. The desired angle θ (α in the figure) is obtained as follows:
θ = 90 o - β.
The sinus appears as a result of applying reduction formulas.
Task example
Let's move on to the practical use of the knowledge gained. We solve a typical problem at an angle between a line and a plane. The following coordinates of the four points are given:
P = (1, -1, 0);
Q = (-1, 2, 2);
M = (0, 3, -1);
N = (-2, -1, 1).
It is known that a plane passes through the points PQM, and a straight line passes through MN. Using the coordinate method, the angle between the plane and the line must be calculated.
First, we write the equations of the line and the plane. For a straight line it is not difficult to compose:
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 this plane. We have:
PQ¯ = (-2, 3, 2);
QM¯ = (1, 1, -3) =>
n¯ = [PQ¯ * QM¯] = (-11, -4, -5).
Now, in the equation of the general plane, we substitute the coordinates of any point lying in it to obtain the value of the free term D:
P = (1, -1, 0);
- (A * x + B * y + C * z) = D =>
D = - (-11 + 4 + 0) = 7.
The equation of the plane has the form:
11 * x + 4 * y + 5 * z - 7 = 0.
It remains to apply the formula for the angle formed when the line and the plane intersect in order to get an answer to the problem. We have:
(u¯ * n¯) = (11, 4, 5) * (- 2, -4, 2) = -28;
| u¯ | = √24; | n¯ | = √162;
θ = arcsin (28 / √ (162 * 24)) = 26.68 o .
Using the example of this problem, we showed how to use the coordinate method to solve geometric problems.