Page 2: Vector Operations and Bases

This page delves deeper into vector operations and introduces the concept of vector bases, which is essential for understanding vector mathematics in higher dimensions.

The page covers:

• Scalar multiplication of vectors in more detail
• The concept of vector bases

Definition: A vector base is a set of linearly independent vectors that can be used to represent any vector in the space.

The page provides examples of vector bases and introduces the canonical basis, which is fundamental in vector algebra.

Example: In 2D space, the canonical basis consists of vectors i = (1,0) and j = (0,1).

The concept of orthogonal and orthonormal bases is introduced, which is crucial for understanding more advanced topics in vector algebra.

Highlight: Orthonormal bases, such as the canonical basis, simplify many calculations in vector mathematics.

The page also covers the properties of vector operations, including:

• Commutativity of vector addition
• Distributivity of scalar multiplication over vector addition

These concepts are essential for students learning about vector operations and preparing for more advanced topics in linear algebra.

Page 3: Dot Product and Vector Projections

This page focuses on the dot product (scalar product) of vectors and its applications, including vector projections and angle calculations. These concepts are crucial for students studying product escalar de dos vectores (dot product of two vectors).

The page covers:

• Definition and calculation of the dot product
• Geometric interpretation of the dot product
• Angle between vectors using the dot product formula

Formula: The dot product of two vectors a and b is given by a · b = |a| |b| cos(θ), where θ is the angle between the vectors.

The page explains how to use the dot product to determine if vectors are perpendicular:

Highlight: If the dot product of two vectors is zero, they are perpendicular (orthogonal) to each other.

Vector projections are introduced, which are important applications of the dot product:

Definition: The projection of a vector v onto a vector u is given by the formula: proj_u v = (v · u / |u|^2) u

The page emphasizes the importance of these concepts in various fields of mathematics and physics, making it essential for students studying vectores 1 bachillerato ejercicios resueltos (vector exercises for high school students).