The scalar product in the plane : 11th grade math lesson

The 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 :

Definition:

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 :

Property 1:

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 :

Property 2:

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

.

II. Scalar product and orthogonality :

Definition:

To say that and are two orthogonal vectors means that :

– Either or ;

– Let (OA)(OB), with and nonzero.

Ownership:

.

III. Properties of the scalar product :

Properties:

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 :

Ownership:

Let and be non-zero.

2. Al-Kashi theorem:

Theorem:

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

We have:

3. Median theorem:

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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