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.