In mathematics, vectors (also known as Euclidean vectors, geometric vectors and vectors) refer to quantities with magnitude and direction. It can be imagined as a line segment with an arrow. The arrow indicates the direction of the vector; Line segment length: indicates the size of the vector. The quantity corresponding to a vector is called a quantity (called a scalar in physics), and a quantity (or scalar) has only a size and no direction.

Vector notation: print letters (such as A, B, U, V) in bold, and add a small arrow "→" at the top of the letter when writing. [ 1]? If the starting point (a) and the ending point (b) of the vector are given, the vector can be recorded as AB (and added to the top →). In the space Cartesian coordinate system, vectors can also be expressed in pairs. For example, (2,3) in the xOy plane is a vector.

In physics and engineering, geometric vectors are more often called vectors. Many physical quantities are vectors, such as the displacement of an object, the force exerted on it by a ball hitting a wall and so on. On the contrary, it is scalar, that is, a quantity with only size and no direction. Some definitions related to vectors are also closely related to physical concepts. For example, vector potential corresponds to potential energy in physics.

The concept of geometric vector is abstracted in linear algebra, and a more general concept of vector is obtained. Here, a vector is defined as an element of a vector space. It should be noted that these abstract vectors are not necessarily represented by number pairs, and the concepts of size and direction are not necessarily applicable. Therefore, it is necessary to distinguish the concept of "vector" in the text according to the context when reading on weekdays. However, we can still find the basis of a vector space to set the coordinate system, and we can also define the norm and inner product on the vector space by choosing a suitable definition, which enables us to compare abstract vectors with specific geometric vectors.

Algebraic representation

General printing is represented by lowercase English letters (a, b, c, etc.). ) bold, while handwriting is indicated by adding an arrow (→) before the letters A, B and C, for example

, can also be represented by capital letter AB, CD(→ with arrow (→), etc.

Geometric representation

A vector can be represented by a directed line segment. The length of the directed line segment indicates the size of the vector, the size of the vector, that is, the length of the vector. A vector of length 0 is called a zero vector, and a vector of length 1 unit is called a unit vector.

Vector representation

The direction indicated by the arrow indicates the direction of the vector. [ 1]?

Coordinate representation

In the plane rectangular coordinate system, two unit vectors I and J with the same direction of X axis and Y axis are taken as a set of bases. A is an arbitrary vector in a plane rectangular coordinate system, with the coordinate origin O as the starting point and P as the end point. According to the basic theorem of plane vector, there is only one pair of real numbers (x, y), which makes a=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 a point.

The coordinates of. Vector a is called the position vector of point p [1]?

Coordinate representation of vector

In the spatial rectangular coordinate system, three unit vectors I, J and K in the same direction as the X-axis, Y-axis and Z-axis are taken as a set of bases. If it is an arbitrary vector in the coordinate system, take the coordinate origin O as the starting point to make the vector A. According to the basic theorem of space, there is only one set of real numbers (x, y, z), which makes a=ix+jy+kz. Therefore, the pair of real numbers (x, y, z) is called the coordinate of vector A, and is denoted as a=(x, y, z). This is the coordinate representation of vector A, where (x, y, z) is the coordinate of point P, and vector A is called the position vector of point P..

Of course, for multi-dimensional space vectors, it can be obtained by analogy.

Matrix representation of vectors

directed line segment

Specified line segment

From the finish line,

Is the end point, then the line segment has a starting point.

from beginning to end

The direction and length of.

A line segment with direction and length is called a directed line segment. [ 1]?

Modulus of vector vector

The size of the vector is the length (or modulus) of the vector. The modulus of vector a is expressed as |a|. [ 1]?

note:

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

. [ 1]?

2. Because the direction can't compare with the size, the vector can't compare with the size. The concepts of "greater than" and "less than" are meaningless to vectors. take for example

Meaningless.

Vector unit vector

A vector whose length is one unit (that is, the modulus is 1) is called a unit vector. A vector with a length of 1 that is the same as that in the direction A is called the unit vector in the direction A, and is denoted as

. [ 1]?

Vector negative vector

If the modulus of vector AB and vector CD are equal and the directions are opposite, then we call vector AB the negative vector of vector CD, which is also called the inverse vector. [ 1]?

Vector zero vector

A vector with a length of 0 is called a zero vector and recorded as 0. The starting point and ending point of the zero vector coincide, so the zero vector has no definite direction, or the direction of the zero vector is arbitrary. [ 1]?

Vector 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. [ 1]?

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.

Vector free vector

A vector whose starting point is not fixed can move in parallel at will. After moving,

vectors

The 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.

Vector sliding vector

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

Vector fixed vector

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

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.

Vector direction vector

The vector A on the straight line L and the vector line with vector a*** are called the direction vector on the straight line L.

Vector opposite vector

The vector with the same length and opposite direction as A is called the inverse quantity of A, and it is denoted as -a, with -(-A) = A, and the inverse quantity of zero vector is still zero vector. [ 1]?

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 (* * * line), which is represented as A ∨ B. The length of zero vector is zero, and it is a vector with the same starting point and ending point, and its 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. [ 1]?

If a=(x, y) and b=(m, n), then a//b→a×b=xn-ym=0.

Vector * * * vector oriented

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.

Note: Only three or more vectors can speak * * * face, not * * * face.

Vector normal vector

Take the direction vector A of the straight line l⊥α, and the vector A is called

normal vector

Normal vector of plane α. [2]?

Modulus of vector sum

Let points A(x 1, y 1) and B(x2, y2) in the plane rectangular coordinate system xOy, then

set up

vector addition

The addition of vectors satisfies parallelogram rule and triangle rule,

Vector addition

Algorithm of vector addition;

Exchange law: a+b = b+a;

Law of association: (a+b)+c=a+(b+c).

Vector subtraction

If a and b are mutually opposite vectors, then the reciprocal of a=-b, b=-a and a+b =0. 0 is 0.

OA-OB=BA。 That is, "* * * has the same starting point and points to the quilt.

Vector subtraction

Minus sign "

A=(x 1, y 1) and b=(x2, y2), then a-b=(x 1-x2, y 1-y2).

As shown in the figure: c=a-b starts with B and ends with A. ..

Law of addition and subtraction transformation: a+(-b)=a-b

Vector multiplication

The cross product of real number λ and vector A is a vector, denoted as λ a, and |λa|=|λ|*|a|. [ 1]?

When λ >; 0, the direction of λa is the same as that of A; When λ

Note: By definition, if λa=0, then λ=0 or A = 0.

Real number λ is called the coefficient of vector A, and the geometric meaning of multiplier vector λa is to extend or compress the directed line segment representing vector A. ..

When |λ| > 1, the directed line segment of vector a is in the original direction (λ >; 0) or the opposite direction (λ

When | λ|; 0) or the opposite direction (λ

The dot product of real number p and vector a is a number.

The multiplication of numbers and vectors satisfies the following algorithm.

Law of association: (λ a) b = λ (a b) = (a λ b).

The distribution law of vector logarithm (first distribution law): (λ+μ)a=λa+μa 。

The distribution law of number pair vector (second distribution law): λ(a+b)=λa+λb 。

The elimination method of number multiplication vector: ① If the real number λ≠0 and λa=λb, then A = B. ② If a≠0 and λa=μa, then λ = μ.

It should be noted that the addition, subtraction, multiplication and division of vectors (vectors without division) satisfy the real number addition, subtraction, multiplication and division algorithm.

Vector quantity product

Definition: Given two non-zero vectors A and B, let OA = A and OB = B, then ∠AOB is called the included angle between vector A and vector B, denoted as θ, and specified as 0≤θ≤π.

Definition: the product of two vectors (inner product, dot product) is a quantity (undirected), which is recorded as A B.

If a and b are not lines, then

; If a, b***, then

. [ 1]?

The coordinates of the product of vectors are expressed as: a b = x x'+y y'.

Vector product algorithm

A b = b a (commutative law)

(λ a) b = λ (a b) (on the associative law of number multiplication)

(a+b) c = a c+b c (distribution law)

Properties of scalar product of vectors

A a = the square of a |.

a⊥b〈=〉a b=0 .

| a b |≤| a ||| b |. (The formula is proved as follows: | a b | = | a || b||| cos α| Because 0≤|cosα|≤ 1, | AB |≤| A |||||| B |)

The main difference between vector product and real number operation

The product of 1. vector does not satisfy the associative law, that is, (a b) c ≠ a (b c); For example: (a b)? ≠a？ b？ .

2. The product of vectors does not satisfy the law of elimination, that is, b=c cannot be deduced from A = A = C (A ≠ 0).

| ab | and | a | b | are not equivalent.

4. From |a|=|b|, a=b or a=-b cannot be deduced, but vice versa.

Vector cross product

Definition: the cross product of two vectors a and b.

Geometric representation of vectors

(Outer product, cross product) is a vector marked as a×b (here × is not a multiplication symbol, but a representation, which is different from "∧"). If a and b are not * * * lines, the modulus of a×b is: ∣ A× B ∣ = | A || B | SIN < A, b >；; The direction of a×b is perpendicular to A and B, and A, B and a×b form a right-handed system in this order. If A and B are vertical, then ∣a×b∣=|a|*|b| (this is different from the product of quantities, please note). If a×b=0, then A and B are parallel. The cross product is a set of normal vectors on the plane where two non-zero vectors of a non-zero line lie.