Orthogonality and equations of lines: 1st grade math course to download in PDF.

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 (O,\vec{i},\vec{j}).

1. directing vector of a line.

Definition:

The directing vector of a line (d) is any representative of the vector \vec{AB} where

A and B are any two distinct points on the line (d).

Example:

In the image below, the vectors \vec{AB}(2;1), \vec{u}(-2;-1) and \vec{v}(4;2) are vectors

directors of the line (d).

direction vector

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 M(x;y)

of a line verify a relation ax+by+c=0 where a, b and c are real numbers.

Demonstration:

Let P(x_P;y_P) and Q(x_Q;y_Q) be two points of a line (d).

Then, for any point M(x;y) belonging to (d), we have :

the vectors \vec{PM}(x-x_P;y-y_P) and \vec{PQ}(x_Q-x_P;y_Q-y_P) are collinear.

So we have det(\vec{PM};\vec{PQ})=0.

That is (x-x_P)(y_Q-y_P)-(y-y_P)(x_Q-x_P)=0.

x(y_Q-y_P)-x_P(y_Q-y_P)-y(x_Q-x_P)+y_P(x_Q-x_P)=0

so (y_Q-y_P)x+(x_P-x_Q)y+(y_Px_Q-x_Py_Q)=0.

By posing a=y_Q-y_P; b=x_P-x_Q and c=y_Px_Q-x_Py_Q,

we have the equation of the line (d) which is of the form ax+by+c=0.

Definition:

The relationship ax+by+c=0 is called a Cartesian equation of the line (d).

Ownership:

The vector \vec{u}(-b;a) is a directing vector of the line (d) of Cartesian equation ax+by+c=0.

Example:

If the Cartesian equation of the line (d) is 5x+4y-11=0, then the vector \vec{u}(-4;5)

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 ax+by+c=0 and a'x+b'y+c'=0 where a,b,c,a',b',c' are real numbers.

The lines (d) and (d’) are parallel if, and only if, ab'-a'b\neq0.

Proof:

The director vectors of the lines (d) and (d’) are, respectively, \vec{u}(-b;a) and \vec{v}(-b';a').

The lines (d) and (d’) are intersecting if and only if the vectors \vec{u} and \vec{v} are not collinear.

In other words, if the determinant of these two vectors is non-zero.

That is, -ba'-(a(-b'))=-ba'+ab'=ab'-a'b\neq0.

2. Secant lines and systems of equations

Theorem:

When two lines intersect, the coordinates (x;y) of their point of intersection

are solution of the system :

system of equations

3. Perpendicular lines

Theorem:

Let be two lines (d) and (d’) with respective Cartesian equations ax+by+c=0 and a'x+b'y+c'=0 where a,b,c,a',b',c' are real numbers.

The director vectors of the lines (d) and (d’) are, respectively, \vec{u}(-b;a) and \vec{v}(-b';a').

The lines (d) and (d’) are perpendicular if, and only if, \vec{u}.\vec{v}=0 is aa'+bb'=0.

Proof:

The director vectors of the lines (d) and (d’) are, respectively, \vec{u}(-b;a) and \vec{v}(-b';a').

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. :

\vec{u}.\vec{v}=0

This is equivalent to :

-b\times  \,(-b')+a\times  \,a'=0 or aa'+bb'=0.

Cette publication est également disponible en : Français (French) العربية (Arabic)

Télécharger puis imprimer cette fiche en PDF

Télécharger ou imprimer cette fiche «orthogonality and equations of lines: 1st grade math course to download in PDF.» au format PDF afin de pouvoir travailler en totale autonomie.


D'autres fiches dans la section 1st grade math class




Télécharger nos applications gratuites Mathématiques Web avec tous les cours,exercices corrigés.

Application Mathématiques Web sur Google Play Store.    Application Mathématiques Web sur Apple Store.    Suivez-nous sur YouTube.


D'autres articles analogues à orthogonality and equations of lines: 1st grade math course to download in PDF.


  • 96
    Exponential function: 1st grade math course download PDF.In mathematics, the exponential function is the function that is noted exp and is equal to its own derivative and takes the value 1 in 0. Moreover, it is used to model phenomena in which a constant difference on the variable leads to a constant ratio. The student will need…
  • 95
    Equations and inequalities of the second degree: 1st grade math course in PDFYou can download this free course on second degree math equations and inequations in PDF format. In mathematics, a second degree equation is a polynomial equation of degree 2. In this said equation, x is the unknown and the letters a, b, c represent the coefficients, with a being different…
  • 95
    Probability: 1st grade math course to download in PDF.This math course on probability is available for free download in pdf format. I. Calculating frequencies Definition: In a bivariate statistical series, the values are usually represented in a cross tabulation. The sums of the rows and columns of a double entry table are called the table margins. They appear…
Les dernières fiches mises à jour

Voici la liste des derniers cours et exercices ajoutés au site ou mis à jour et similaire à orthogonality and equations of lines: 1st grade math course to download in PDF. .

  1. Integrals : corrected high school math exercises in PDF.
  2. Intégrales : exercices de maths en terminale corrigés en PDF.
  3. الوظائف والحدود: تمارين الرياضيات في السنة النهائية مصححة بتنسيق PDF.
  4. Functions and limits: senior math exercises corrected in PDF.
  5. Fonctions et limites : exercices de maths en terminale corrigés en PDF.


Inscription gratuite à Mathématiques Web. Mathématiques Web c'est 2 146 112 fiches de cours et d'exercices téléchargées.

Copyright © 2008 - 2023 Mathématiques Web Tous droits réservés | Mentions légales | Signaler une Erreur | Contact

.
Scroll to Top
Mathématiques Web

FREE
VIEW