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.

Definition:

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).

Application and Method:
1. We calculate the coordinates of a directing vector of the line.
2. 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.

1. Determine a direction vector of the line (BC).
2. Detail the construction of the parallel to (BC) passing through A.

### 2. Cartesian equation of the right.

Theorem:

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 .

Definition:

The relationship is called a Cartesian equation of the line (d).

Ownership:

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

Theorem:

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

Theorem:

When two lines intersect, the coordinates of their point of intersection

are solution of the system :

### 3. Perpendicular lines

Theorem:

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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