# 6 - Vectors

If you need to, read again the 8th Grade lesson on the vectors. It contains the basics to be able to handle this 9th Grade lesson.

## Collinear vectors

 are collinear are not collinear

Two vectors are collinear if they have the same direction, that means if you draw lines on them, they'll be parallel. In that case you can find one of the two vectors by multiplying the other by some number k : if two vectors and are collinear, then there is a number k such as . On the contrary, you can also say that if there is a number k such as then the two vectors are collinear. In the case of the drawing above (on the left), k = - 2.

## Collinearity and coordinates

You can also express the collinearity of two vectors by using their coordinates : actually, if and , then .
As , the coordinates of and k are the same, hence :
Hence , and
To conclude : Two vectors and are collinear if

To put into practice :
The collinearity of vectors can be used to prove lots of things in geometry. For example, if you want to prove that 3 points in the plane are in the same line, then you can prove that the vectors that pass through these points are collinear.

Question :
In an orthonormal frame, are the points A (-2 ; -1), B (6 ; 3), and C (9 ; 5) in a line ?
To determine it, let's calculate the coordinates of the vectors and and see if they are collinear.

then, the vectors and are not collinear, that means the points A, B, and C are not on a line.

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