Table of Contents
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
3 ways to express vectors
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
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.
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.
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
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:
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"
Baidu Encyclopedia and Wikipedia
In mathematics, vectors (also called Euclidean vectors, geometric vectors, vectors) refer to quantities with magnitude and direction. It can be visualized as a line segment with an arrow. Pointed by the arrow: represents the direction of the vector; line length: represents the size of the vector. The quantity corresponding to a vector is called quantity (called scalar in physics), and quantity (or scalar) has only size and no direction.
The notation of vectors: the letters (such as a, b, u, v) written in bold (bold) in printed form, and a small arrow "→" is added to the top of the letter when writing.If the start point (A) and end point (B) of a vector are given, the vector can be recorded as AB (and add → on top).In the spatial rectangular coordinate system, vectors can also be expressed in pairs, for example, (2,3) in the xOy plane is a vector.
A vector space (also called a linear space) is a vector called a collection of objects, which can be added together and multiplied by a number ("scaling"), a so-called scalar.Scalars are usually considered real numbers, but there are also vector spaces where scalars are multiplied by complex numbers, rational numbers, or generally any field.The operations of vector addition and scalar multiplication must meet certain requirements listed below, called axioms.
Euclidean vectors are an example of a vector space. They represent physical quantities, such as forces: any two forces (of the same type) can be added to produce a multiplicative force vector of a third sum. A real multiplier is another force vector. Similarly, but in a more geometric sense, vectors that represent displacements in a flat or three-dimensional space also form a vector space. Vectors in vector space do not necessarily have to be arrow-like objects, as they appear in the example above: vectors are considered abstract mathematical objects with specific properties, and in some cases can be considered 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 the space. Infinite-dimensional vector space appears naturally in mathematical analysis. As a function space, its vector is a function. These vector spaces often have additional structures, which can be topological structures, allowing for issues of proximity and continuity. Among these topologies, a topology defined by a specification or inner product is more commonly used because it has the concept of distance between two vectors. Especially Banach space and HilbertThe situation in space is the basis of mathematical analysis.