What is a vector? What is called a vector

In mathematics, quantities that have both magnitude and direction are called vectors (the same as vectors, with no beginning or end).

This is called an n-dimensional vector. where ai is called the i-th component of the vector.

a1"of"1"is a subscript for a"ai"of"i"is a subscript, and by analogy) In C++, there are also vectors.

1. Algebraic representation: generally printed with bold lowercase letters, or a, b, c

etc. Vector representation.

Handwriting is used in a, b, c...and so on by adding an arrow to indicate it.

2 Geometric representation: Vectors can be represented by directed line segments. The length of the directed segment represents the magnitude of the vector, and the direction of the arrow indicates the direction of the vector.

If the endpoint A of the line segment AB is the starting point and B is the end point, then the line segment has the direction and length from the starting point A to the end point B.

Geometric representation of vectors.

This type of line segment with direction and length is called a directed line segment. ）

1 Definition of a vector: A quantity that has both magnitude and direction is called a vector. Such as force, displacement, velocity, etc. in physics.

Vectors can be represented by the letters a, b, c, etc., or they can be represented by the start and end letters of the directed line segments that represent the vector (the starting point is written in front, the end point is written in the back, and arrows are drawn on it).

2 Vector modulus: The size of the vector ab (i.e., the length of the vector ab) is called the modulus of the vector ab.

The modulo of a vector is a non-negative real number, a scalar quantity with magnitude and no direction.

3 Concepts of zero vectors, unit vectors, parallel vectors, collinear vectors, and equal vectors.

1) Zero vector: A vector with zero length (modulus) is called a zero vector, which is denoted as 0

The direction of the zero vector can be regarded as arbitrary, and the zero vector is specified to be parallel to either vector.

2) Unit vector: A vector with a length (modulus) of 1 unit length is called a unit vector.

3) Parallel vectors: Non-zero vectors with the same or opposite direction are called parallel rows.

Because any set of parallel vectors can be moved to the same straight line, parallel vectors are also called collinear vectors.

4) Equality vectors: Vectors of equal length and the same direction are called equality vectors.

A vector is a quantity that has a magnitude and direction. It can be visualized as a line segment with an arrow. The arrows point in the direction of the vector; The length of the line segment represents 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) is only a magnitude and has no direction.

The concept of geometric vectors is abstracted in algebra to obtain a more general vector concept of preparation. Vectors are defined as elements of a vector space, and it is important to note that these abstract vectors are not necessarily represented by pairs, nor do the concepts of size and direction apply. Therefore, when reading on a daily basis, it is necessary to distinguish what is said in the text according to the context"Vectors"What kind of concept is it?

However, it is still possible to find the basis of a vector space to set up the coordinate system, and it is also possible to mediate the norm and inner product on the vector space by choosing an appropriate definition of imitation block, which allows us to analogize vectors in the sense of equilibrium to specific geometric vectors.

A vector is a quantity that has a magnitude and direction. It can be visualized as a line segment with an arrow. The arrows point in the direction of the vector; The length of the line segment represents the size of the vector.

The quantity corresponding to the vector is called the quantity (called the scalar in the theory of the Martial Arts), and the quantity (or scalar) is only a magnitude and has no direction.

The concept of geometric vectors is abstracted in Liangzhi linear algebra to obtain a more general concept of vectors. Vectors are defined as elements of vector space, and it should be noted that these abstract vectors are not necessarily represented in pairs, and the concepts of size and direction are not necessarily applicable. Therefore, when reading on a daily basis, it is necessary to distinguish what is said in the text according to the context"Vectors"What kind of concept is it?

However, it is still possible to find the basis of a vector space to set up the coordinate system, and it is also possible to mediate the norm and inner product on the vector space by choosing the appropriate definition, which allows us to analogy vectors in the abstract sense to concrete geometric vectors.

Expand; Computation of vectors.

Associativity: (a)·b= (a·b)=(a· b).

The distributive property of a vector for a number (the first distributive property) :( a= a+ a

The distributive property of numbers for vectors (second distributive property): a+b) = a+ b.

The elimination law of the number multiplication vector: If the real number ≠0 and a= b, then a=b. If a≠0 and a= a, then =

The arithmetic of the quantity product of a vector:

a·b = b·a (commutative law).

a)·b= (a·b) (associative property on number multiplication) a+b)·c=a·c+b·c (distributive property).

Vector product of vectors:

a×b=-b×a

a)×b=λ(a×b)=a×(λb)

a×(b+c)=a×b+a×c.

a+b)×c=a×c+b×c.

The coordinate representation of the product of the plane vector quantities is: if a=(x,y), b=(x,y), then a·b=x ·x +y ·y.

Two non-zero vectors a, b, then|, are knowna||b|cos (which is the angle between a and b) is called the product of a and b or the product of a circle. Written as a·b. The quantity product of two vectors is equal to the sum of the products of their corresponding coordinates.

The quantity product a·b is equal to the length of a|a|Projection with b in the direction of a|b|The product of cos.

Vectors

In mathematics, a vector (also known as a Euclidean vector, geometric vector, vector), refers to a quantity with magnitude and direction. It can be visualized as a line segment with an arrow. The arrow points to:

represents the direction of the vector; The length of the line segment represents the size of the vector. The quantity corresponding to the vector is called the quantity (called the scalar in physics), and the quantity Wang Chun (or scalar) only has a magnitude and no direction.

In physics and engineering, geometric vectors are more commonly referred to as vectors. Many physical quantities are vectors, such as the displacement of an object, the force exerted on a ball as it hits a wall, and so on. The opposite is a scalar quantity, which is a quantity that has only a magnitude and no direction.

Some vector-related definitions are also closely related to physical concepts, such as the potential of a vector corresponding to potential energy in physics.

The above content reference:Encyclopedia – Vector

The components of the vector.

Elements similar to the Nian Dan matrix (a vector can also be understood as a matrix of a row or a column), such as (a1, a2, a3) This vector has three components: a1, a2, a3. where AI is called the i-th component. The number of components is called the dimension of the vector.

The number of vectors.

This is a word in a vector group (some row vectors of the same dimension, or some columns of the same dimension) that refers to the number of vectors in the group.

Number of rows, number of columns.

It is the concept of a matrix, which corresponds to a vector, and should be a matrix of vector groups, that is, for row vector groups, each vector is used as a row of the matrix to form a matrix, similar to a column.

The dimension of a vector, as mentioned earlier, the number of vector components is called the dimension of the vector.

The number of dimensions of the vector space.

If there are r vectors linearly independent, and any vector in the linear space can be linearly represented by these r vectors, r is called the dimension of the vector space, and the r vectors are called the basis of the space.

The rank of the vector group.

If there are r vectors that are linearly independent, and any vector in the vector group can be linearly represented by these r vectors, r is said to be the dimension of the vector space, and r vectors are called to be extremely independent of the vector group.

Finally correct 1 mistake for you.

The original vector is irrelevant, and it remains irrelevant after adding components.

It should be: the original vector group is linear independent, and the vector group after adding components to each vector at the same position is still irrelevant.

Similar to the latter sentence, it is also wrong.

This has nothing to do with the whole, in fact it can be seen as the limitation and expansion of space.

Vector: A quantity that has both size and direction. In general, they are called vectors in physics and vectors in mathematics. In computers, vector graphics can be infinitely enlarged and never deformed.

Vectors are vectors, and they are called differently. A vector is a quantity with a direction, and a scalar is a quantity with only quantity and no direction.

Vectors In mathematics, quantities that have both magnitude and direction and follow the parallelogram rule are called vectors, which are different from vectors in physics, which only have directions and magnitudes, and have no starting point (also known as free vectors).

In mathematics, quantities that have only size but no direction are called quantities, and in physics they are often called scalars. For example, distance.

Vector Some physical quantities require both numerical magnitude (including the relevant units) and direction to be completely determined. Operations between these quantities do not follow general algebraic rules, but rather special ones. For example, a physical quantity such as displacement is called a physical vector.

Some physical quantities only have numerical magnitude (including the relevant units) and are not directional. Operations between these quantities follow general algebraic laws. For example, physical quantities such as temperature and mass are called physical scalar quantities.

Quantities with magnitude and direction are called vectors (also known as vectors).

A quantity that has both magnitude (length) and direction is called a vector;

Vectors with the same or opposite direction are parallel vectors; Also called collinear vectors; (In high school, it is stipulated that zero vectors are parallel vectors to all vectors).

Vectors with the same direction and equal length are equal vectors;

As mentioned in these concept textbooks, a parallel vector is two vectors that are parallel, which is similar to the parallel of two straight lines, which can be understood by analogy.

An equality vector is a parallel vector, but the two vectors must be in the same direction to be equal.

Collinear vectors are parallel vectors, and collinear vectors include parallel vectors.

The concept of vectors.

A quantity that has both direction and size is called direction.

Quantities (called vectors in physics), and quantities that have no direction in size are called quantities (called scalars in physics).

Geometric representation of vectors.

A directional line segment is called a directed line segment, and a directed line segment with A as the starting point and B as the end point is denoted as AB. (ab is the printed type, and the writing type is the top plus one).

The length of the directed line segment ab is called the modulo of the vector and is denoted as |ab|。

A directed line segment consists of three factors: the starting point, the direction, and the length.

A vector whose length is equal to 0 is called a zero vector and is denoted as 0. The direction of the zero vector is arbitrary; A vector whose length is equal to 1 unit of length is called a unit vector.

Equality vectors vs. collinear vectors.

Vectors of equal length and the same direction are called equal vectors.

Two non-zero vectors with the same or opposite direction are called parallel vectors, vectors a and b are parallel and denoted as a b, zero vectors are parallel to any vector, i.e., 0 a, and parallel vectors are also called collinear vectors.

Vector operations.

Additive operations. ab bc ac, this calculation rule is called the triangle rule of vector addition.

It is known that two vectors oa and ob start from the same point o, and oa and ob are adjacent sides to make a parallelogram oacb, then the diagonal oc starting from o is the sum of vectors oa and ob, and this calculation rule is called the parallelogram rule of vector addition.

For zero vectors and arbitrary vectors a, there are: 0 a 0 a.

a＋b|≤|a|＋|b|。

The addition of vectors satisfies all the laws of addition.

Subtraction. A vector equal in length and in the opposite direction is called the opposite vector of a, (a) a, the opposite vector of the zero vector is still a zero vector.

1）a＋(－a)＝(a)＋a＝0（2）a－b＝a＋(－b)。

Multiplication of numbers. The product of the real number and the vector a is a vector quantity, and this operation is called the number multiplication of the vector, denoted as a, |λa|＝|a|when

At 0, the direction of a is the same as the direction of a when

At 0, the direction of a is opposite to the direction of a, when

0, a=0.

Let , be a real number, then: (1)( a

(μa)（2）(λ

)a=λaa（3）λ(a±b)

a±λb（4）(－a

(λa)λ(a)。

The addition, subtraction, and multiplication operations of vectors are collectively referred to as linear operations.

The quantitative product of the vector.

Two non-zero vectors a, b, then|, are knowna||b|cos

It is called the product of quantities or the inner product of a and b, denoted as a b, and is the angle between a and b, |a|cos

|b|cos

This is called the projection of the vector a in the b direction (b in the a direction). The product of the quantities of zero vectors and arbitrary vectors is 0.

The geometric meaning of a b: the quantity product a b is equal to the length of a|a|Projection with b in the direction of a|b|cos

The product of . The quantity product of two vectors is equal to the sum of the products of their corresponding coordinates.

A parallel vector is a collinear vector, that is, a vector in the same or opposite direction, a vector on a straight line.

Equality vectors are equal on the basis that parallel vectors are collinear vectors, and the loud modulus is also equal, that is, they are all equal in magnitude and direction (basically the same vector).