Instruction

1

To determine

**the angle**between two**straight**in space even if they do not intersect. In this case, you need to mentally combine the start of their directing vectors and calculate the magnitude of the resulting angle. In other words, is any of the adjacent angles formed by intersecting**straight**drawn parallel data.2

There are several ways to specify a straight line in space, for example, vector-parametric, parametric and canonical. The three mentioned method is convenient to use when finding the angle, because all of them require the introduction of coordinates of directing vectors. Knowing these values, you can define an educated

**angle**cosine theorem of vector algebra.3

Suppose two lines L1 and L2 is set to the canonical equations:L1: (x – x1)/k1 = (y – y1)/l1 = (z – z1)/n1;L2: (x – x2)/k2 = (y – y2)/l2 = (z – z2)/n2.

4

Using the values of ki, li and ni, write down the coordinates of directing vectors of the straight. Call them N1 and N2:N1 = (k1, l1, n1);N2 = (k2, l2, n2).

5

The formula for the cosine of the angle between the vectors represents the ratio between their dot product and the result of arithmetic multiplication of their lengths (modules).

6

Define the scalar product as the sum of the products of their x, y and applicator that con:N1•N2 = k1•k2 + l1•l2 + n1•n2.

7

Calculate the square roots of the sums of the squares of the coordinates to define the modules of the guides of the vectors:|N1| = √(k12 + l12 + n12);|N2| = √(k22 + l22 + n22).

8

Use all the expressions to write the General formula of the cosine of the angle N1N2:cos (N1N2) = (k1•k2 + l1•l2 + n1•n2)/( √(k12 + l12 + n12)•√(k22 + l22 + n22)).To find the value of the angle, calculate the arccos of this expression.

9

Example: determine

**the angle**between the given**straight**:L1: (x - 4)/1 = (y + 1)/(-4) = z/1;L2: x/2 = (y - 3)/(-2) = (z + 4)/(-1).10

Solution:N1 = (1, -4, 1); N2 = (2, -2, -1).N1•N2 = 2 + 8 – 1 = 9;|N1|•|N2| = 9•√2.cos (N1N2) = 1/√2 → N1N2 = π/4.

# Advice 2: How to find the angle between the crossing straight lines

To determine the magnitude of the angle between the crossing straight lines, you need both a direct (or one of them) moved to a new position by parallel transfer to the intersection. You should then find the measure of the angle between the intersecting lines.

You will need

- Ruler, triangle, pencil, protractor.

Instruction

1

Modern technologies in various industries (construction, engineering, instrumentation, etc.) are based on the construction of three-dimensional (three-dimensional) models. The basis of such a construction is a three-dimensional design (in the school course the solution to spatial problems is considered in the section of geometry called solid geometry). Quite often in three-dimensional design required the solution of problems of quantification of the mutual arrangement of intersecting straights, for example, distances and angles between them.

2

Under crossed lines defined as straight lines which do not belong to the same plane. The angle between two straight lines that do not belong to the same plane, is equal to the value of the angle between two intersecting straight lines, respectively, parallel data skew lines.

3

Therefore, to determine the angle between two straight lines that do not belong to the same plane, it is necessary to arrange them parallel straight lines in the same plane, that is, to reduce the problem to finding the angle between two intersecting straight (viewed in plane geometry).

4

It is absolutely equitable are three variants of straight lines in space:

-a direct parallel to the first straight line, is conducted through any point of the second straight line;

-a direct parallel to the second straight line, is conducted through any point of the first straight;

-straight and parallel first and second straight, are conducted through arbitrary point of space.

-a direct parallel to the first straight line, is conducted through any point of the second straight line;

-a direct parallel to the second straight line, is conducted through any point of the first straight;

-straight and parallel first and second straight, are conducted through arbitrary point of space.

5

If two intersecting lines form two pairs of supplementary angles. The angle between two intersecting straight is considered the smaller of the adjacent angles, formed by the intersection of the straight lines (called adjacent angles whose sum is 180°). The measurement of the angle between intersecting straight leads to the solution of the problem of the magnitude of the angle between the crossed lines.

6

For example, the two belong to different planes a and b direct. On one of the lines, for example, choose a arbitrary point a, through which when using the ruler and a right triangle are holding direct b' such that b' || b. According to the theorem of parallel transport, the magnitude of angles in this type of spatial displacement is a constant. Thus, a video forms with parallel lines b and b' equal angles. Using the protractor measure the angle between intersecting straight a and b'.

Note

You should observe the accuracy of geometric constructions and measurement angle.

Useful advice

A better option is to build a direct parallel to one of the data direct, through any point of the second straight line.