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Course on vectors and translation, we will review the location in the plane and the coordinates in an orthonormal frame as well as the coordinates of a vector.
At the end of this lesson, the student should have acquired the following skills:
- Know how to calculate the length of a segment in an orthonormal frame of reference;
- Know how to determine the coordinates of the middle of a segment;
- Know how to determine if two vectors are equal with or without coordinates;
- Know how to determine, request, assign a value and display a variable in an algorithm.
- Middle of a segment;
- Distance between two points or norm of a vector of the plane;
- Equality of vectors (coordinates, parallelogram, vectors and midpoint).
I. Notion of vector and translation
1.Vector translation 
Let A and B be two points in the plane.
The translation that transforms A into B associates to any point of the plane C the point D such that the segments [AD] and [BC] have the same middle.
It is called the translation of vector , often noted
.
Remark:
The quadrilateral ABDC is then a parallelogram, possibly flattened.
Construct the image of point C and that of point N by the translation of vector .
2. Equal vectors
Two vectors and
are equal if the translation that transforms A into B also transforms C into D.
We note .
Two vectors and are equal if and only if the quadrilateral ABDC is a parallelogram, possibly flattened.
3.Representative of a vector
The translation of vector also transforms C into D, E into F.
We have .
They are representatives of the same vector, which can be noted for example.
4. specific vectors
The null vector, associated with the translation that transforms A into A, B into B, C into C….
We have
The vector opposite to the vector is the vector associated with the translation that
transforms B into A: this is the vector .
We have .
The point I is the middle of the segment [AB], if and only if, .
II. Coordinates in an orthonormal reference frame of the plane
In an orthonormal reference frame of the plane , consider a vector
and M the image of point O by the translation of vector
.
1. definition and properties
The coordinates of the vector are the coordinates of the point M such that :
.
We note or
.
Remark:
The null vector has coordinates .
Ownership:
Two vectors are equal if and only if they have the same coordinates in the same reference frame.
2.coordinates of a vector in the plane
In an orthonormal plane, let A and B be the points of coordinates and
.
The coordinates of the vector coordinates of are
.
3.norm of a vector.
The norm of a vector is the length of the vector
which is noted
.
In an orthonormal plane :
If then
.
Remark:
This equality comes from the Pythagorean theorem.
4. Distance between two points or length of a segment
In an orthonormal reference frame of the plane.
If and
then
.
5. coordinates of the middle of a segment
The point I is the middle of the segment [AB] has coordinates :
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