# Vectors and location in the plane and translation: course in 2nd grade in PDF.

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 $\vec{AB}$

Definition:

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

Definition:

Two vectors and are equal if the translation that transforms A into B also transforms C into D.

We note .

Ownership:

Two vectors and are equal if and only if the quadrilateral ABDC is a parallelogram, possibly flattened.

### 3.Representative of a vector

Definition:

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

Definitions:

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 .

Definition of the middle of a segment :

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

Definition:

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

Definition:

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.

Definition:

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

Ownership:

In an orthonormal reference frame of the plane.

If and then .

### 5. coordinates of the middle of a segment

Ownership:

The point I is the middle of the segment [AB] has coordinates :

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