Topic: the origin of vectors and vector symbols write a paper.

Definition of quantity

In mathematics, a quantity with only size but no direction is called a quantity (or scalar), and it is often called a scalar in physics.

Definition of vector

A quantity with both magnitude and direction is called a vector.

Note: A vector in linear algebra refers to an ordered array of n real numbers, which is called an n-dimensional vector. α=(a 1, a2, …, an) is called n-dimensional vector, where ai is called the i-th component of vector α.

("1" of "a 1" is the subscript of A, "I" of "ai" is the subscript of A, and so on).

[Edit this paragraph] Representation of vectors

1. Algebraic representation: Generally, printing is represented by letters α, β, γ … or A, B, c …. Handwriting is indicated by adding an arrow to the letters a, b, c….

2. Geometric representation: vectors can be represented by directed line segments. The length of the directed line segment indicates the size of the vector, and the direction pointed by the arrow indicates the direction of the vector. (If endpoint A of line segment AB is defined as the starting point and endpoint B is defined as the ending point, then the line segment has the direction and length from the starting point A to the ending point B ... This line segment with direction and length is called a directed line segment. )

3. Coordinate representation: in the plane rectangular coordinate system, two unit vectors I and J in the same direction as the X axis and the Y axis are respectively taken as the base. A is an arbitrary vector in a plane rectangular coordinate system, starting from the coordinate origin O, and the vector OP = A. According to the basic theorem of plane vectors, there are only one pair of real numbers (x, y), so that a= vector OP=xi+yj. Therefore, the pair of real numbers (x, y) is called the coordinate of vector A, and is denoted as a=(x, y). This is the coordinate representation of vector A, where (x, y) is the coordinate of point P, and vector OP is called the position vector of point P..

[Edit this paragraph] Modulus of vectors and number of vectors.

The size of the vector is the length (or modulus) of the vector. The modulus of vector a is expressed as |a|.

note:

1, the modulus of the vector is a non-negative real number, and the sizes can be compared.

2. Because directions can't compare sizes, vectors can't compare sizes. The concepts of "greater than" and "less than" are meaningless to vectors. For example, "vector ab >；; Vector CD "is meaningless.

[Edit this paragraph] Special vector

unit vector

A vector with a length of 1 is called a unit vector. A vector in the same direction as the vector A with the length of 1 is called the unit vector in the direction A, and is denoted as a0, where a0=a/|a|.

Zero vector

A vector with a length of 0 is called a zero vector, and its starting point and ending point coincide, so the zero vector has no definite direction, or the direction of the zero vector is arbitrary.

Equal vector

Vectors with the same length and direction are called equal vectors. Vectors a and b are equal, so let's say a = B.

Rule: All zero vectors are equal.

When the vector is represented by a directed line segment, the starting point can be arbitrarily selected. Any two equal nonzero vectors can be represented by the same directed line segment, regardless of the starting point of the directed line segment. Directed line segments with the same direction and length all represent the same vector.

Free vector

A vector whose starting point is not fixed can move in parallel at will, and the moved vector still represents the original vector.

In the sense of free vector, equal vectors are regarded as the same vector.

Only free vectors are studied in mathematics.

Sliding vector

A vector acting along a straight line is called a sliding vector.

Fixed vector

A vector acting on a point is called a fixed vector (also called a viscous vector).

Position vector

For any point p on the coordinate plane, we call the vector OP the position vector of the point p, and write it as: vector p.

[Edit this paragraph] The opposite vector

The vector with the same length and opposite direction as A is called the inverse quantity of A, which is -A .. Yes-(-a) = a;

The inverse of a zero vector is still a zero vector.

Parallel vector

Non-zero vectors with the same or opposite directions are called parallel lines (or * * * lines) vectors. Vectors A and B are parallel (* * * lines), denoted as A ∨ B. 。

The length of zero vector is zero, the starting point and the ending point coincide, and the direction is uncertain. We stipulate that the zero vector is parallel to any vector.

A set of vectors parallel to the same line is a * * * line vector.

* * * quantity orientation

Three (or more) vectors parallel to the same plane are called * * * vectors.

The vector in space has and only has the following two positional relationships: (1) * * * plane; (2) not * * * face.

Only three or more vectors can talk about * * * surfaces, not to mention * * * surfaces.

[Edit this paragraph] Vector operation

Let a=(x, y) and b=(x', y').

1, vector addition

The addition of vectors satisfies parallelogram rule and triangle rule.

AB+BC=AC .

a+b=(x+x '，y+y ').

a+0=0+a=a .