Vector Norms and Scalar Multiplication

This page focuses on the concepts of vector norms and scalar multiplication, essential topics in Cours complet sur les vecteurs PDF.

The section begins by defining the norm of a vector as its length, introducing the notation ||u|| for the norm of vector u.

Definition: The norm of a vector is its length, denoted by ||u|| for a vector u.

An example is provided to illustrate the calculation of vector norms in a coordinate system.

Example: For a vector u = (3, 4), its norm is calculated as ||u|| = √(3² + 4²) = 5.

The concept of scalar multiplication of vectors is then introduced, explaining how multiplying a vector by a real number affects its magnitude and direction.

Highlight: The product of a vector u by a real number k is denoted as ku and results in a vector with the same or opposite direction as u, depending on whether k is positive or negative.

The properties of scalar multiplication are explored, including its effect on vector direction and magnitude.

Vocabulary: Collinear vectors are vectors that have the same or opposite directions.

The page concludes with an important theorem relating scalar multiplication to the midpoint of a line segment.

Quote: "The point K is the midpoint of [AB] if and only if AK = 1/2 AB."

This section provides a solid foundation for understanding vector operations and their geometric interpretations, crucial for mastering Vecteurs maths seconde exercices and Formules vecteurs Seconde.