Instruction

1

Video is one of the fundamental geometric concepts along with the point and the plane. It's an endless shape, which you can connect any two points in space. Video always belongs to any plane. Based on the location of two straight lines, one should use different methods of finding distance between them.

2

There are three options for the location of two straight lines in space relative to each other: they are parallel, intersect or cross each other. The second option is possible only if they lie in the same plane, the first does not exclude facilities two parallel planes. The third situation suggests that direct lie in different parallel planes.

3

To find the distance between two parallel lines, it is necessary to determine the length of the perpendicular segment connecting any two points. Since direct have two identical coordinates, which follows from the definition of their parallelism, the line equation in two-dimensional coordinate space can be written as:

L1: a•x + b•y + C = 0;

L2: a•x + b•y + d = 0.

If you can find a segment length using the formula:

s = |s - d|/√(a2 + b2), and it is easy to see that if C = D, i.e., direct matching, the distance will be zero.

L1: a•x + b•y + C = 0;

L2: a•x + b•y + d = 0.

If you can find a segment length using the formula:

s = |s - d|/√(a2 + b2), and it is easy to see that if C = D, i.e., direct matching, the distance will be zero.

4

It is clear that the distance between intersecting straight lines in two-dimensional coordinate system has no meaning. But when they are located in different planes, it is possible to find the length of a line lying in a plane perpendicular to both of them. The ends of this segment will be the point being the projection of any two points direct to the surface. In other words, its length equal to the distance between parallel planes containing these lines. Thus, if the plane is set to the General equations:

α: A1•x + B1•y + C1•z + E = 0,

β: A2•x + B2•y + C2•z + F = 0,

distance between lines can be calculated by the formula:

s = |E – F|/√(|A1•A2| + B1•B2 + C1•C2).

α: A1•x + B1•y + C1•z + E = 0,

β: A2•x + B2•y + C2•z + F = 0,

distance between lines can be calculated by the formula:

s = |E – F|/√(|A1•A2| + B1•B2 + C1•C2).

Note

In General, direct and skew in particular is interesting not only for mathematicians. Their properties are useful in many other applications: in building and architecture, in medicine and in nature.

# Advice 2 : How to find the distance between straight lines in space

To calculate the distance between straight lines in three-dimensional space, you need to determine the length of a line belonging to a plane perpendicular to both of them. Such a calculation makes sense if they are crossed, i.e. are in two parallel planes.

Instruction

1

Geometry is the science that finds application in many areas of life. It would be unthinkable to design and build ancient, old and modern buildings, without its methods. One of the simplest geometric figures is video. A combination of several such shapes forms a spatial surface based on their relative position.

2

In particular, straight lines are in different parallel planes, can interbreed. The distance at which they are located from each other can be represented in the form of a perpendicular line segment lying in a corresponding plane. The ends of this limited area will be a direct projection of two points of intersecting lines on its plane.

3

You can find the distance between straight lines in space as the distance between the planes. Thus, if they are given by equations of the General form:

β: A•x + B•y + C•z + F = 0,

γ: A2•x + B2•y + C2•z + G = 0, then the distance is determined by the formula:

d = |F - G|/√(|A•A2| + |B•B2| + |•C2|).

β: A•x + B•y + C•z + F = 0,

γ: A2•x + B2•y + C2•z + G = 0, then the distance is determined by the formula:

d = |F - G|/√(|A•A2| + |B•B2| + |•C2|).

4

The coefficients a, A2, B, B2, C and C2 are the coordinates of the normal vectors of these planes. Since skew lines lie in parallel planes, then these values should be correlated with each other in the following proportion:

A/A2 = B/B2 = C/C2, i.e. they either are equal or differ by the same multiplier.

A/A2 = B/B2 = C/C2, i.e. they either are equal or differ by the same multiplier.

5

Example: let the two plane 2•x + 4•y – 3•z + 10 = 0 and -3•x – 6•y + 4,5•z – 7 = 0 that contain the skew lines L1 and L2. Find the distance between them.

Solution.

These planes are parallel because their normal vectors are collinear. This is evidenced by the equality:

2/-3 = 4/-6 = -3/4,5 = -2/3, where -2/3 is the multiplier.

Solution.

These planes are parallel because their normal vectors are collinear. This is evidenced by the equality:

2/-3 = 4/-6 = -3/4,5 = -2/3, where -2/3 is the multiplier.

6

Divide the first equation for the multiplier:

-3•x – 6•y + 4,5•z – 15 = 0.

Then the formula of the distance between the lines is converted into the following form:

d = |F - G|/√(A2 + B2 + C2) = 8/√(9 + 36 + 81/4) ≈ 1.

-3•x – 6•y + 4,5•z – 15 = 0.

Then the formula of the distance between the lines is converted into the following form:

d = |F - G|/√(A2 + B2 + C2) = 8/√(9 + 36 + 81/4) ≈ 1.