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The equation is an important element in mathematics. Its good mastery will allow you to progress throughout the year.
I. Directing vectors and Cartesian equation
In this chapter, we place ourselves in an orthonormal reference frame .
1. directing vector of a line.
The directing vector of a line (d) is any representative of the vector where
A and B are any two distinct points on the line (d).
Example:
In the image below, the vectors ,
and
are vectors
directors of the line (d).
- We calculate the coordinates of a directing vector of the line.
- The line (BC) and its parallel have the same direction vectors, it is enough to take a representative of origin A.
Example:
Let be three points A(1;5), B(-3;2) and C(2;-1) in an orthonormal frame.
- Determine a direction vector of the line (BC).
- Detail the construction of the parallel to (BC) passing through A.
2. Cartesian equation of the right.
In an orthonormal reference frame, the coordinates of the set of points
of a line verify a relation where a, b and c are real numbers.
Demonstration:
Let and
be two points of a line (d).
Then, for any point belonging to (d), we have :
the vectors and
are collinear.
So we have .
That is .
so .
By posing ;
and
,
we have the equation of the line (d) which is of the form .
The relationship is called a Cartesian equation of the line (d).
The vector is a directing vector of the line (d) of Cartesian equation
.
Example:
If the Cartesian equation of the line (d) is , then the vector
is a directing vector of this line.
II. Relative positions of lines
1. parallel or secant lines
Let be two lines (d) and (d’) with respective Cartesian equations and
where
are real numbers.
The lines (d) and (d’) are parallel if, and only if, .
Proof:
The director vectors of the lines (d) and (d’) are, respectively, and
.
The lines (d) and (d’) are intersecting if and only if the vectors and
are not collinear.
In other words, if the determinant of these two vectors is non-zero.
That is, .
2. Secant lines and systems of equations
When two lines intersect, the coordinates of their point of intersection
are solution of the system :
3. Perpendicular lines
Let be two lines (d) and (d’) with respective Cartesian equations and
where
are real numbers.
The director vectors of the lines (d) and (d’) are, respectively, and
.
The lines (d) and (d’) are perpendicular if, and only if, is
.
Proof:
The director vectors of the lines (d) and (d’) are, respectively, and
.
The lines are perpendicular if, and only if, these two direction vectors are orthogonal.
This means that the scalar product of these two vectors is zero, i.e. :
This is equivalent to :
or
.
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