One of the axioms of geometry claims that through any two points it is possible to draw a single line. This axiom indicates that there is a single numerical expression that uniquely describes the specified one-dimensional geometric object. We consider in the article the question of how to make an equation for a line passing through two points.
What is a point and a straight line?
Before considering the question of constructing in space and on the plane of a direct equation passing through a pair of different points, it is necessary to define these geometric objects.
A point is uniquely determined by a set of coordinates in a given system of coordinate axes. Apart from them, there are no longer characteristics for a point. It is a zero-dimensional object.
When talking about a straight line, then each person represents the line depicted on a white sheet of paper. At the same time, an exact geometric definition of this object can be given. A straight line is such a collection of points for which connecting each of them with all the others will give a set of parallel vectors.
This definition is used when defining the vector equation of the line, which will be discussed below.
Since a line of arbitrary length can be noted on any line, they say that it is a one-dimensional geometric object.
Vector numeric function
The equation through two points of the passing line can be composed in different forms. In three-dimensional and two-dimensional spaces, the main and intuitive numerical expression is vector.
Suppose that there is some directed segment u¯ (a; b; c). In three-dimensional space, the vector u¯ can begin at an arbitrary point; therefore, its coordinates specify an infinite set of parallel vectors. However, if we select a specific point P (x 0 ; y 0 ; z 0 ) and put it at the beginning of the vector u¯, then multiplying this vector by an arbitrary real number λ, we can obtain all points in the space in one straight line. That is, the vector equation is written in the form:
(x; y; z) = (x 0 ; y 0 ; z 0 ) + λ * (a; b; c)
Obviously, for the case on the plane, the numerical function takes the form:
(x; y) = (x 0 ; y 0 ) + λ * (a; b)
The advantage of this type of equation over the others (in segments, canonical, general) is that it explicitly contains the coordinates of the guide vector. The latter is often used to determine whether the lines are parallel or perpendicular.
Common in segments and canonical function for a line in two-dimensional space
When solving problems, sometimes it is necessary to write an equation of a straight line passing through two points in a specific, concrete form. Therefore, other ways of specifying this geometric object in two-dimensional space should be given (for simplicity, we consider the case on the plane).
We start with a general equation. It has the form:
A * x + B * y + C = 0
As a rule, on the plane the equation of the line is written in this form, only y is explicitly defined through x.
Now transform the expression above as follows:
A * x + B * y = -C =>
x / (- C / A) + y / (- C / B) = 1
This expression is called the equation in segments, since the denominator for each variable shows how long the line segment cuts off on the corresponding coordinate axis relative to the starting point (0; 0).
It remains to give an example of the canonical equation. To do this, we write explicitly the vector equality:
x = x 0 + λ * a;
y = y 0 + λ * b
From here we express the parameter λ and equate the resulting equalities:
λ = (x - x 0 ) / a;
λ = (y - y 0 ) / b;
(x - x 0 ) / a = (y - y 0 ) / b
The last equality is called an equation in canonical or symmetric form.
Each of them can be converted to vector and vice versa.
The equation of a straight line through two points passing through: a compilation technique
We return to the question of the article. Suppose that in space there are two points:
M (x 1 ; y 1 ; z 1 ) and N (x 2 ; y 2 ; z 2 )
A single straight line passes through them, the equation of which is very simple to compose in vector form. To do this, calculate the coordinates of the directed segment MN¯, we have:
MN¯ = N - M = (x 2 -x 1 ; y 2 -y 1 ; z 2 -z 1 )
It is not difficult to guess that this vector will be a guide for the line, the equation of which must be obtained. Knowing that it also passes through M and N, you can use the coordinates of any of them for a vector expression. Then the desired equation takes the form:
(x; y; z) = M + λ * MN¯ =>
(x; y; z) = (x 1 ; y 1 ; z 1 ) + λ * (x 2 -x 1 ; y 2 -y 1 ; z 2 -z 1 )
For the case in two-dimensional space, we obtain a similar equality without the participation of the variable z.
Once the vector equality for the line is written, it can be translated into any other form that requires a problem question.
Task: to compose a general equation
It is known that a straight line passes through points with coordinates (-1; 4) and (3; 2). It is necessary to compose the equation of the line passing through them, in a general form, expressing y through x.
To solve the problem, we first write the equation in vector form. The coordinates of the vector (guide) are equal to:
(3; 2) - (-1; 4) = (4; -2)
Then the vector form of writing the equation of the line is as follows:
(x; y) = (-1; 4) + λ * (4; -2)
It remains to write it in general form in the form y (x). We rewrite this equality explicitly, express the parameter λ and exclude it from the equation:
x = -1 + 4 * λ => λ = (x + 1) / 4;
y = 4 - 2 * λ => λ = (4-y) / 2;
(x + 1) / 4 = (4-y) / 2
From the obtained canonical equation, we express y and come to the answer to the question of the problem:
y = -0.5 * x + 3.5
The validity of this equality can be verified by substituting the coordinates of the points specified in the condition of the problem.
Task: a line passing through the center of the segment
Now we will solve one interesting problem. Suppose that two points M (2; 1) and N (5; 0) are given. It is known that a straight line passes through the middle of the segment that connects the points and is perpendicular to it. Write the equation of the line passing through the middle of the segment in vector form.
The desired numerical expression can be compiled by calculating the coordinate of this center and determining the direction vector, which with a segment makes an angle of 90 o .
The coordinate of the middle of the segment is:
S = (M + N) / 2 = (3.5; 0.5)
Now we calculate the coordinates of the vector MN¯:
MN¯ = N - M = (3; -1)
Since the directing vector for the line is perpendicular to MN¯, their scalar product is equal to zero. This allows you to calculate the unknown coordinates (a; b) of the guide vector:
a * 3 - b = 0 =>
b = 3 * a
Now we write the vector equation:
(x; y) = (3,5; 0,5) + λ * (a; 3 * a) =>
(x; y) = (3.5; 0.5) + β * (1; 3)
Here we replaced the product a * λ with the new parameter β.
Thus, we have compiled the equation of a line passing through the center of the segment.