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**scalar product**in the plane in a

**11th grade math lesson**where we will study the definition and the different properties of the scalar product. In this

**eleventh grade lesson**, we will see the relationship between the scalar product and the concept of orthogonality. Then, we will finish with applications with the cosine of an angle, the Al-Kashi theorem and the median theorem.

## I. Different expressions of the scalar product :

### 1. Colinear vectors :

be and two non-zero collinear vectors, such that

and .

– If and have the same meaning: .

– If and are in opposite directions: .

– If or then .

– is the scalar square of the vector

### 2. Any vectors :

Let and be two non-zero vectors such that

and .

So:

.

A’ and B’ are respectively the orthogonal projects of A on (OB) and of B on (OA).

### 3. Properties :

Let (x;y) and (x’;y’) be the respective coordinates of the vectors and in any **orthonormal reference frame**.

.

## II. Scalar product and orthogonality :

To say that and are two orthogonal vectors means that :

– Either or ;

– Let (OA)(OB), with and nonzero.

### 2. Property:

.

## III. Properties of the scalar product :

### Properties:

Let be three vectors and k a real number.

– (symmetry).

– (linearity)

– (linearity)

– (linearity)

– (remarkable identity)

– (remarkable identity)

– (remarkable identity)

## IV. Applications of the scalar product:

### 1. Scalar product and cosine :

Let and be non-zero.

### 2. Al-Kashi theorem:

Let ABC be a triangle such that AB=c, AC=b and BC=a.

We have:

### 3. Median theorem:

Let A and B be two distinct points and I the middle of the segment [AB].

For any point M, :

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