Line and plane are the two most important geometric elements that can be used to construct different shapes in 2D and 3D space. Consider how to find the distance between parallel lines and parallel planes.
Math task straight line
From the school geometry course it is known that in a two-dimensional rectangular coordinate system a line can be specified in the following form:
y=kx + b.
Where k and b are numbers (parameters). The written form of a line in a plane is a plane that is parallel to the z-axis in three-dimensional space. In view of this, in this article, for the mathematical assignment of a straight line, we will use a more convenient and universal form - a vector one.
Assume that our line is parallel to some vector u¯(a, b, c) and passes through the point P(x0, y0, z0). In this case, in vector form, its equation will be represented as follows:
(x, y, z)=(x0, y 0, z0) + λ(a, b, c).
Here λ is any number. If we explicitly represent the coordinates by expanding the written expression, then we will get a parametric form of writing a straight line.
It is convenient to work with a vector equation when solving various problems in which it is necessary to determine the distance between parallel lines.
Lines and the distance between them
It makes sense to talk about the distance between lines only when they are parallel (in the three-dimensional case, there is also a non-zero distance between skew lines). If the lines intersect, then it is obvious that they are at zero distance from each other.
The distance between parallel lines is the length of the perpendicular connecting them. To determine this indicator, it is enough to choose an arbitrary point on one of the lines and drop a perpendicular from it to another.
Let's briefly describe the procedure for finding the desired distance. Suppose that we know the vector equations of two lines, which are presented in the following general form:
(x, y, z)=P + λu¯;
(x, y, z)=Q + βv¯.
Construct a parallelogram on these lines so that one of the sides is PQ, and the other, for example, u. Obviously, the height of this figure, drawn from the point P, is the length of the required perpendicular. To find it, you can apply the following simpleformula:
d=|[PQ¯u¯]|/|u¯|.
Since the distance between straight lines is the length of the perpendicular segment between them, then according to the written expression, it is enough to find the modulus of the vector product of PQ¯ and u¯ and divide the result by the length of the vector u¯.
An example of a task to determine the distance between straight lines
Two straight lines are given by the following vector equations:
(x, y, z)=(2, 3, -1) + λ(-2, 1, 3);
(x, y, z)=(1, 1, 1) + β(2, -1, -3).
From the written expressions it is clear that we have two parallel lines. Indeed, if we multiply by -1 the coordinates of the direction vector of the first line, we get the coordinates of the direction vector of the second line, which indicates their parallelism.
Calculate the distance between straight lines using the formula written in the previous paragraph of the article. We have:
P(2, 3, -1), Q(1, 1, 1)=>PQ¯=(-1, -2, 2);
u¯=(-2, 1, 3).
Then we get:
|u¯|=√14cm;
d=|[PQ¯u¯]|/|u¯|=√(90/14)=2.535 cm.
Note that instead of points P and Q, absolutely any points that belong to these lines could be used to solve the problem. In this case, we would get the same distance d.
Setting a plane in geometry
The question of the distance between the lines was discussed above in detail. Now let's show how to find the distance between parallel planes.
Everyone represents what a plane is. According to the mathematical definition, the specified geometric element is a collection of points. Moreover, if you compose all possible vectors using these points, then all of them will be perpendicular to one single vector. The latter is usually called the normal to the plane.
To specify the equation of a plane in three-dimensional space, the general form of the equation is most often used. It looks like this:
Ax + By + Cz + D=0.
Where capital Latin letters are some numbers. It is convenient to use this kind of plane equation because the coordinates of the normal vector are explicitly given in it. They are A, B, C.
It is easy to understand that two planes are parallel only when their normals are parallel.
How to find the distance between two parallel planes ?
To determine the specified distance, you should clearly understand what is at stake. The distance between planes that are parallel to each other is understood as the length of the segment perpendicular to them. The ends of this segment belong to planes.
The algorithm for solving such problems is simple. To do this, you need to find the coordinates of absolutely any point that belongs to one of the two planes. Then, you should use this formula:
d=|Ax0+ By0+Cz0+ D|/√(A2 + B2 + C2).
Since the distance is a positive value, the modulus sign is in the numerator. The written formula is universal, since it allows you to calculate the distance from the plane to absolutely any geometric element. It is enough to know the coordinates of one point of this element.
For the sake of completeness, we note that if the normals of two planes are not parallel to each other, then such planes will intersect. The distance between them will then be zero.
The problem of determining the distance between planes
It is known that two planes are given by the following expressions:
y/5 + x/(-3) + z/1=1;
-x + 3/5y + 3z – 2=0.
It is necessary to prove that the planes are parallel, and also to determine the distance between them.
To answer the first part of the problem, you need to bring the first equation to a general form. Note that it is given in the so-called form of an equation in segments. Multiply its left and right parts by 15 and move all terms to one side of the equation, we get:
-5x + 3y + 15z – 15=0.
Let's write out the coordinates of two normal vectors of the planes:
1¯=(-5, 3, 15);
2¯=(-1, 3/5, 3).
It can be seen that if n2¯ is multiplied by 5, then we will exactly get the coordinates n1¯. Thus, the considered planes areparallel.
To calculate the distance between parallel planes, select an arbitrary point of the first of them and use the above formula. For example, let's take the point (0, 0, 1) which belongs to the first plane. Then we get:
d=|Ax0+ By0+ Cz0 + D|/√(A2 + B2 + C2)=
=1/(√(1 + 9/25 + 9))=0.31 cm.
Desired distance is 31 mm.
Distance between plane and line
The theoretical knowledge provided also allows us to solve the problem of determining the distance between a straight line and a plane. It has already been mentioned above that the formula that is valid for calculations between planes is universal. It can also be used to solve the problem. To do this, just select any point that belongs to the given line.
The main problem in determining the distance between the considered geometric elements is the proof of their parallelism (if not, then d=0). Parallelism is easy to prove if you calculate the scalar product of the normal and the direction vector for the line. If the elements under consideration are parallel, then this product will be equal to zero.
