A vector to understand vector

What is a vector?

Vectors have two main dimensions: size and direction.

Size: The length of the arrow indicates the size

Direction: The direction pointed by the arrow indicates the direction

Vector has 2 dimensions of size and direction

3 ways to express vectors

Algebraic representation

Lowercase English letters for general printing (a,b,cEtc.), handwriting is indicated by adding an arrow (→) to the letters a, b, and c, such as

Algebraic representation of vectors


Geometric representation

Vectors can be represented by directed line segments. The length of the directed line segment represents the size of the vector, and the size of the vector, that is, the length of the vector.

Geometric representation of a vector


Coordinate representation

In the plane rectangular coordinate system, take two unit vectors in the same direction as the x-axis and y-axis, respectively.i.jAs a set of substrates. a is an arbitrary vector in the plane rectangular coordinate system, and the vector a is set with the coordinate origin O as the starting point P as the end point. From the basic theorem of the plane vector, we know that there is only one pair of real numbers (x, y) such that a = xi + yj, so the pair of real numbers (x, y) is called the coordinates of the vector a, and is written as a = (x, y) . This is the coordinate representation of the vector a. Where (x, y) is the coordinates of point P. The vector a is called the position vector of the point P.

Coordinate representation of vector

In the space rectangular coordinate system, take 3 unit vectors that are the same as the x-axis, y-axis, and z-axis directions, respectively.i.j.kAs a set of substrates. If it is an arbitrary vector in the coordinate system, the vector a is set with the coordinate origin O as a starting point. According to the basic theorem of space, there is only one set of real numbers (x, y, z), so that a = ix + jy + kz, so the real number pair (x, y, z) is called the coordinate of the vector a, and is written as a = (x, y, z). This is the coordinate representation of the vector a. Where (x, y, z) are the coordinates of point P. The vector a is called the position vector of the point P.

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

Matrix representation of vectors

Matrix representation of vectors


Relationship of scalar, vector, matrix, tensor

These 4 concepts are constantly rising in dimensionality. It is easier to understand the metaphorical explanation with the concept of point-line polyhedron:

  • Point - scalar (Scalar
  • Line - vector
  • Face-matrixmatrix
  • Body - tensor

Relationship of scalar, vector, matrix, tensor

Interested parties can learn more about the following:

'Understanding scalars in one article"

'Understanding Vectors in One Article"

'Understanding matrices in one article"

'Understand tensor in one article"

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Baidu Encyclopedia and Wikipedia

Baidu Encyclopedia version

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

The notation of the vector: the printed body is recorded as a bold (bold) letter (such as a, b, u, v), and a small arrow "→" is added to the top of the letter when writing. If the start (A) and end (B) of the vector are given, the vector can be denoted as AB (and added to the top →). In a space Cartesian coordinate system, vectors can also be represented in pairs, for example, xOy plane (2,3) is a vector.

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Wikipedia version

Vector spaces (also known as linear spaces) are vectors that are called collections of objects, which can be added together and multiplied by numbers ("zooms"), so-called scalars. A scalar is usually considered a real number, but there is also a vector space where the scalar is multiplied by a complex number, a rational number, or usually any field. Vector addition and scalar multiplication operations must satisfy certain of the requirements listed below, called axioms.

Euclidean vectors are an example of vector space. They represent physical quantities, such as forces: any two forces (of the same type) can be added to produce a third sum of multiplicative force vectors from one real multiplier to another force vector. Again, but in a more geometric sense, vectors representing displacements in a plane or three-dimensional space also form a vector space. Vectors in vector space do not have to be arrow-shaped objects because they appear in the above example: vectors are treated as abstract mathematical objects with specific properties, which in some cases can be treated as arrows.

Vector spaces are the subject of linear algebra and are well characterized by their dimensions. Roughly speaking, it specifies the number of independent directions in space. The infinite dimensional vector space naturally appears in mathematical analysis. As a function space, its vector is a function. These vector spaces typically have an additional structure, which can be a topology, allowing for consideration of proximity and continuity issues. In these topologies, the topology defined by the specification or inner product is more commonly used because it has a distance between the two vectors. Especially the Banach space and HilbertIn the case of space, this is the basis of mathematical analysis.

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