A straight line is a basic geometric element for constructing more complex shapes in space and on a plane. In two-dimensional space, lines can either intersect or be parallel. In the three-dimensional case, crossing lines are still added to them. In this article, we consider the definition of a line parallel to another line or plane.
What is a straight line?
When they say: "Define the lines of parallel parallel," you must clearly understand what kind of geometric element in question. By a straight line is meant a collection of points in which all vectors formed by two arbitrary points are parallel. This definition can be given differently: a line is a line that connects two given points with a segment of the smallest length.
The figure above shows how the two points A and B are connected by the smallest line segment.
Any line has a direction, which is defined by its directing vector, and is a one-dimensional object. The latter fact means that you can measure the length of the segments - parts of the line.
Equation for line
If a certain coordinate system is specified, then one mathematical equality can be used to write down the set of all points that form a given line. This equality is called the equation of the line. It can be recorded in several ways. Here we consider only three of them.
If the directing vector of the line u¯ (a; b; c) in three-dimensional space and some point M (x 0 ; y 0 ; z 0 ) lying on the line are known, then its equation is written as follows:
(x; y; z) = (x 0 ; y 0 ; z 0 ) + λ * (a; b; c).
Here λ (lambda) is a parameter that can take any number. In fact, this equation takes the point M to any other point on the line using the vectors λ * u¯. This equation is called a vector equation.
Opening the vector equation, we come to the parametric expression:
x = x 0 + λ * a;
y = y 0 + λ * b;
z = z 0 + λ * c.
For the two-dimensional case, we have similar expressions with two coordinates:
(x; y) = (x 0 ; y 0 ) + λ * (a; b) and
x = x 0 + λ * a;
y = y 0 + λ * b.
For a straight line on a plane, consider another way of defining it. To do this, we express the lambda parameter in the last type of equation and equate the resulting equalities:
(xx 0 ) / a = (yy 0 ) / b =>
A * x + B * y + C = 0, where A = 1 / a; B = -1 / b; C = y 0 / b - x 0 / a.
The resulting equation is called the general equation of a line in the plane. It can be rewritten in a more familiar form:
y = k * x + p, where k = -A / B; p = -C.
Parallel lines
The definition of parallel lines is easiest to give using the concept of a vector guide. Two lines will be parallel only if their direction vectors are those. This definition is valid in any spaces.
Parallel segments can also be used to define straight parallel lines. So, if two arbitrary segments, each of which belongs to the corresponding line, are parallel, then such lines will be.
Vector parallelism can be checked in two ways:
- Calculate the cosine of the angle between them using the scalar product. It should be equal to 1, which corresponds to an angle of 0 o between the considered geometric elements.
- Represent one vector as a multiplication by a certain number of the second. If this can be done, then the vectors and the lines defined by them will be parallel.
Distance between lines
If two lines intersect, then the distance between them is zero. What definition of the distance between parallel lines can be given? The distance between parallel lines is the length of the vector, the end of which lies on one line and the beginning on the other, while the vector must be perpendicular to both lines.
You can calculate this distance in two ways:
- Solve a system of three equations. The first of these will be the scalar product of the perpendicular vector to any of the directing vectors of the lines. Two other equations of the system are obtained by substituting the coordinates of a perpendicular vector into equations for lines.
- Use the well-known formula, which is obtained from considering the area of the parallelogram. This formula is given below:
d = | [P 1 P 2 ¯ * u¯] | / | u¯ |.
Here u¯ is the vector directing the 1st straight line, P 1 and P 2 are arbitrary points on the 1st and 2nd straight lines, respectively, P 1 P 2 ¯ is the vector built on these points. Please note that in the numerator of the formula is the module of the vector product.
The second method allows you to calculate the distance, however, if a specific point on the line is known, and it is necessary to find a perpendicular vector to the second line with the beginning at a known point, then the first solution method should be applied.
Plane and straight
This is a spatial case. There are only three possible locations for these geometric objects:
- they intersect at one point;
- they do not intersect at any point, which is the definition of a line parallel to the plane;
- all points of the line belong to the plane, that is, they are parallel and the line lies in the plane.
The parallelism of a straight line and a plane is determined from the condition that the product of their directional vectors scalar to zero. If the plane is specified as follows:
A * x + B * y + C * z + D = 0,
then its direction vector has the coordinates n¯ (A; B; C). Then the parallelism condition can be written as follows:
(n¯ * u¯) = 0.
Through a specific point in space that does not belong to a plane, you can draw an infinite number of lines that will be parallel to this plane.
The problem with two lines
Two lines on the plane are described by the following equations:
r1: (x; y) = (1; 0) + λ * (5; 2);
r2: (x; y) = (3; -4) + α * (2; 0.8).
It is necessary to find out whether they are parallel, and find the distance between them.
Proving parallelism is simple. To do this, multiply by 2.5 the second guide vector. We get:
2.5 * (2; 0.8) = (5; 2).
Since we got the first direction vector, then the lines are parallel.
To calculate the distance between them, we choose an arbitrary point on the first line, for example, (1; 0). It will be the beginning of the perpendicular vector. Find the coordinates of its end (x; y) on the second line. The following equations can be written:
(x-1) * 5 + y * 2 = 0;
x = 3+ 2 * α;
y = -4 + 0.8 * α.
We find α:
(3+ 2 * α-1) * 5 + (-4 + 0.8 * α) * 2 = 0 =>
α ≈ -0.1724.
Then the coordinates of the end of the perpendicular vector are equal:
(2.6552; -4.13792).
Then the distance between the lines is:
d = √ ((2,6552-1) 2 + (- 4,13792-0) 2 ) ≈ 4.4567.
The same result can be obtained by using the formula given in the article.