The basic geometric elements are point, line and plane. They are called so because many objects can be built from them, for example, such as a pyramid or a prism. To understand the properties of these figures, it is important to know the relative position in the space of lines and planes. Let's consider this question in more detail in the article.
Definition and description of a point, line and plane
A point in geometry is a 0-dimensional object, the only characteristic of which is its coordinates. The latter are a set of numbers tied to a specific system. For example, on the plane it consists of two elements, in three-dimensional space - of three.
A straight line is a one-dimensional object that has a certain direction. If you connect it to any two points, you get a vector that characterizes it. To describe the lines, several types of equations are used, which can be converted into each other using simple mathematical operations. Here we give only the vector, which is often used to analyze the relative position in the space of lines. It takes the form for a three-dimensional case:
(x; y; z) = (x 0 ; y 0 ; z 0 ) + α * (a; b; c)
Elements with zero indices correspond to some point, which is part of the line. The coordinates that are multiplied by the parameter α (alpha) describe its direction vector along which it passes. Substituting arbitrary numbers α, we can find all the points that form a straight line in space.
Obviously, for a vector equation in two-dimensional space, it is necessary to use only two coordinates for points and vectors.
A plane is a collection of points. The vectors formed on them are perpendicular to a certain direction specified by a vector normal to the plane. All this can be described in several ways. Nevertheless, to solve problems of determining the relative position of a plane and a line, it is convenient to use a general equation. It is written below:
A * x + B * y + C * z + D = 0
The convenience of this form of writing is that the coefficients A, B, C are the coordinates of the perpendicular vector n¯ to the plane.
When solving problems, it is important to consider in which space the problem is being solved. So, the reduced form of the plane equation in the two-dimensional case without the z coordinate will correspond to the equation of the line.
Location of point and line
The relative position of these objects does not depend on whether they are considered on a plane or in space. The definition criteria are constantly the same.
A relatively straight point can only be in two possible positions:
- to lie on her;
- either do not belong to her.
Determining the location option in a specific task is fairly easy. To do this, substitute the coordinates of the desired object in the equation that defines the line. If equality holds, then the point belongs to the line. Otherwise, it is not part of it.
Two lines on the plane
What can be the mutual arrangement of two lines on the plane? There are three different options:
- They intersect at some point.
- They are parallel. That is, they do not intersect at any point.
- They coincide with each other. That is, they intersect at all points.
To understand what the mutual arrangement of lines in a particular case, it is necessary to conduct some mathematical analysis. The following describes the main ideas that should be used in its implementation.
If the direction vectors of the lines are parallel to each other, then the lines, at least, will be parallel. The parallelism of vectors is proved if one of them can be represented as the other multiplied by a real number.
If the guiding vectors are parallel, and at least one point of one line corresponds to another line, then we are talking about completely coincident lines.
If the guiding vectors are not parallel, then the lines intersect at one point. You can find its coordinates by solving a system of equations (these coordinates must correspond to both equations of lines).
A special case of intersection of lines is the angle of intersection equal to 90 o . In this case, they say about the perpendicularity between the considered objects. If two lines are perpendicular, then the scalar product of their guide vectors will be zero.
Line and circle on the plane
Since this object often appears in geometric problems, it is also useful to consider the relative position of the circle and the line. The following options are possible:
- A straight line does not cross a circle.
- She is tangent. That is, the line intersects the circle at a single point.
- She divides it into two arcs. That is, the line intersects the circle at two points.
You can determine the location of these objects for a specific task using the appropriate equations. For a circle centered at (x 0 ; y 0 ) and a radius R, it has the form:
R 2 = (xx 0 ) 2 + (yy 0 ) 2
The determination of the location option is reduced to solving the quadratic equation.
Two lines in space
Often the question arises of what is the mutual arrangement of lines in three-dimensional space. The same options are possible as described in the previous paragraph, however, one more is added to them. The intersecting lines are not intersected, and are not parallel. More details - below.
To determine whether the considered one-dimensional objects are crossed is also not difficult. First of all, it is necessary to find out that their direction vectors are not parallel. After this, the easiest way to calculate the distance between the lines. If it is equal to zero, then they intersect, if it differs (more or less), then they intersect.
Calculation of the distance is made according to the formula:
d = | [M 1 M 2 ¯ * v¯] | / | v¯ |,
Where:
- v¯ is the directing vector of the first line;
- M 1 M2¯ is a vector constructed on arbitrary points M 1 and M 2 of the first and second straight lines, respectively.
The formula can be directly applied if vector equations of lines are given.
Plane and straight
In this case, we are talking about three-dimensional space. The relative position of the plane and the line is possible the following:
- The line belongs to the plane.
- They are parallel to each other.
- A straight line crosses a plane.
Determining the parallelism of these geometric objects is quite simple. To do this, you need to calculate the scalar product of the normal vector of the plane and the directing vector of the line. The fact that this product is equal to zero is a sufficient condition for parallelism. If, moreover, at least one point belongs to the plane, then the whole line lies in it.
If the scalar product is not equal to zero, then the conclusion is as follows. The line and the plane intersect at one point. A special case is the intersection at right angles. If the directing vector of the line can be represented as the product of the number of the normal vector to the plane, then the line and the plane are perpendicular.
The problem with two lines on the plane
Below are two general equations for straight lines in two-dimensional space:
2 * x - y = 7;
-3 * x + 2 * y = 0.
It is necessary to determine the relative position of the lines.
Since there is a case on the plane, there is no need to reduce these equations to vector form. Solving the problem can be easier if you find the roots of the system from these ones. We have:
2 * x - y = 7 => y = 2 * x - 7;
-3 * x + 2 * y = 0 => -3 * x + 2 * (2 * x - 7) = 0 =>
x = 14; y = 21.
Since the system has a unique solution, it corresponds to the intersection of the lines in question at the point (14; 21).
The problem with two lines in space
Two lines are given that are described by equations:
r1: (x; y; z) = (1; -2; 0) + α * (2; -1; 1);
r2: (x; y; z) = (2; 2; 1) + β * (0; 3; -1).
What is the mutual arrangement of lines in space?
It can be noted that the direction vectors are not parallel (no value of the parameter β can give the direction vector r1). That is, the straight lines either intersect or are crossed.
We calculate the distance between them. To do this, take the point M 1 (1; -2; 0) on r1, and the point M 2 (2; 2; 1) on r2. Then the vector connecting them is equal to:
M 1 M 2 ¯ = (1; 4; 1).
Its vector product with a directing vector for r1 is equal to:
[(1; 4; 1) * (2; -1; 1)] = (5; 1; -9).
Since the length of this vector is nonzero, it means that the distance between the lines will be greater than zero. The latter fact says that they do not have common points and are crossed.