Chapter 1 Basic concepts of vector and coordinate § 1 vector

Chapter 1 vector and coordinate

 

Teaching Purpose of this Chapter: Through this chapter, students can master the concept of vectors and their operations, and master the basic properties, operation rules, and component representation of linear and nonlinear operations, it uses vectors and their operations to establish spatial coordinate systems and solve some geometric problems, laying the foundation for the following chapter to study the properties of spatial graphs using algebra.

Emphasis of this Chapter(1) Basic concepts of vectors and various characterization of relations between vectors;

(2) linear operations, Product Operations, calculation rules, and component representation of vectors.

Teaching difficulties in this Chapter(1) Relationship between vectors and their operations and spatial coordinate systems;

(2) Differences and relationships between the vector's quantity product and the vector product;

(3) applications of vectors and their operations in plane and ry.

Teaching Content in this Chapter:

 

 

§ 1. Basic concepts of Vector

1. Definition:Vectors, such as force, speed, and displacement, have both sizes and directions.
   
2. Indicates:In ry, lines with arrows are used to represent the vector, arrows are used to represent the vector direction, and line lengths represent the vector size. The vector size is also called the touch (length) of the vector ).
   

   A vector whose start point is a and end point is B.
   

   
   Note: For convenience, except in a few cases, we will mark the vector with the start and end letters of the vector. We generally Use lowercase letters in black.A, B, c...... Mark the vector, and use the Greek letters λ, μ, ν ...... The number of tags.
   

    

III,Two special Vectors:

1. Zero vector: the zero vector with a modulo equal to 0. It is referred to as the zero vector and recorded as 0.
   

Note: The zero vector is a unique vector with an indefinite direction.

 

2. bitvector: a vector with a modulo equal to 1 is called the unit vector. In particular, the unit vector in the same direction as the non-0 vector is called as the unit vector.

 

IV,Special relationships between vectors:

1. Parallel (collinearity): VectorAParallel to VectorB, MeaningAThe line is parallelBLine, recordedABytesB, Rule: the zero vector is parallel to any vector,

2. Equal: VectorAEqual to VectorB, Meaning, recordedA=B.

Rule: All null vectors are equal.

Note: whether two vectors are equal depends only on their modulo and direction, but is not related to their position. A vector irrelevant to a position is called a free vector.

3. anti-vector: the inverse vector that is equal to the vector modulus but opposite to the vector.

The inverse vector of the zero vector is still its own.

4. Co-plane vector: A group of vectors parallel to the same plane is called a co-plane vector. It is easy to see. Any two vectors are always co-plane, and the zero vector is co-plane with any co-plane vector.

Note: The vector and quantity should be strictly different:

① The vector cannot compare the size. If it is meaningless;

② Division operations are strictly prohibited for vectors. Such subcategories are not allowed to appear.