math
9th Grade - Vectors

9th Grade lesson

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


colinear vectors
are collinear
vectors
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 vector u and vector v are collinear, then there is a number k such as colinear vectors property. On the contrary, you can also say that if there is a number k such as colinear vectors property 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 coordonees vector u and coordinates vector v, then math.
As condition colinearité vectors, the coordinates of coordinates vector v and k vector u are the same, hence :
demonstration vector colineaires
Hence  math equation, and math equation
To conclude : Two vectors vector u and vector v are collinear if math

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 vector AB coordinates and coordinates du vector AC and see if they are collinear.
coordinates of a vector calculation

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



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9th Grade vectors

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