I have a new understanding of vectors. In the world we live in, vectors can be said to be ubiquitous. After learning the vector, I changed my way of looking at things and thinking. Let’s briefly talk about my understanding after learning the vector in the past few days.
The definition of a vector: a quantity that has both magnitude and direction. It is called a vector in physics.
Unit vector: A vector whose length is equal to one unit.
Generally, the measurement is to determine the unit first, and then measure the length, but in the unit vector, you can measure the length first, and then look at the unit again. It breaks the ordinary calculation method and reverses the order in the measurement, and the unit can be large or small. Once it is confirmed, it cannot be changed. Under fixed conditions, the unit vector is recorded as 1.
Zero vector: A vector whose length is equal to zero.
The length is 0, but there is a direction, but the direction is uncertain.
Vectors with equal direction and length, why are equal
vectors defined by both magnitude and direction. When the direction is the same, the two vectors are collinear vectors, because the vector can move arbitrarily as long as the length and direction remain the same. , So they form a line segment with a direction in one dimension, and the length is the length of the longer vector; when the length of the two vectors is also the same, the two vectors are completely overlapped, which is exactly the same from a mathematical point of view , So the two vectors are equal.
On this issue, the definition of the vector must first be clear, otherwise it will not be able to provide sufficient evidence for the conclusion. In solving other problems, it is also necessary to provide a reliable basis in a rigorous manner. The lack of basis for the conclusion is not only not recognized by most people, but also the deeper calculation of the following vector will also be untenable and become impossible to start. . Two concepts are mentioned here, one is the axiom; the other is the intermediate theorem.
Axiom : In mathematics, the word axiom is used in two related but different meanings-logical axioms and non-logical axioms. In both senses, axioms are the starting point for deriving other propositions. Unlike theorems, an axiom (unless there is redundancy) cannot be derived from other axioms, otherwise it is not the starting point itself, but a certain result that can be derived from the starting point-it can simply be classified as a theorem.
Intermediate theorem : also called theorem, is a statement that is proved to be true after being restricted by logic. Generally speaking, in mathematics, only important or interesting statements are called theorems. Proving theorems is the central activity of mathematics.
Talking about'reason' refers to axioms. Axioms are universally recognized principles. They are so simple that they can't be proved by mathematics. Theorems are statements that can be deduced more rigorously on the basis of axioms that can be verified repeatedly. The meaning of axioms is more important than theorems, just like the roots of a tree must penetrate deeply into the soil to grow luxuriant branches. The theorems appeared only after the axioms. No matter how complicated and difficult to understand the later development theorems, they are also derived from a simple one point by one line. Change is always the same. In the final analysis, everything is evolved from simple affairs, and all kinds of knowledge are the same. Therefore, the understanding of axioms is very important for our study.
High-dimensional thinking-what are the learning content considered as a whole
In the learning of vectors, we realize that there are two aspects to consider when using vectors: length and direction. That is to say, the problem must be considered as a whole, and the conditions must not be lost. The vector is no different from the line segment without considering the direction, just as the result cannot be obtained without considering the length or width of the surface of an object, just like blood spreads all over the body.
Among the knowledge we have learned, there is still a lot of knowledge that needs to be considered as a whole. For example, when we learn English words, we must not only look at the spelling and meaning of the words, but also the phonetic transcription of the words. Lack of phonetic transcription to guide accurate pronunciation, only writing can be done in listening, speaking, reading, and writing. The other three listening, speaking, and reading all need phonetic transcription to participate in. The importance of holistic thinking is highlighted. . Therefore, phonetic transcription, word spelling, and word meaning are indispensable in the study of English words.
Also in the training plan, for example, the order in which we study the self-examination is customized by many seniors who have experienced the self-examination through holistic thinking. Take the first two subjects "Operational Research" and "Management Economics" and later "Network Economy and Business Management", "Software Development Tools" and "Database Materials". There are many knowledge points in economics that are not easy to directly understand. "Soft Engineering" and "Database" also learned about database video and software engineering after a year of study, and then I learned these two subjects after laying a solid foundation.
This kind of holistic thinking pervades the entire learning process. We stand on the shoulders of giants to learn and enjoy the results summarized by the predecessors. Therefore, in order to provide a solid shoulder for future generations, we must absorb the experience of the predecessors and work hard. Build a stronger bridge for future generations.
Vector calculation rule
Vector's triangle rule:
Given the non-zero and non-collinear vectors a and b, take any point A in the plane, make vector AB = vector a, make vector BC = vector b through point B, and connect AC to get vector AC.
Then vector AB + vector BC = Vector AC.
That is, vector a + vector b = vector AC.
∵The figure composed of three vectors is exactly a triangle,
∴this rule is called the triangle rule of vectors.
Extension of the vector triangle rule: in the plane, there are n vectors, connected end to end, and the end of the last vector is connected to the beginning of the first vector, then the last vector (the direction is from the beginning of the first vector to the last The end of the vector) is the sum of n vectors.
Parallelogram rule of
vector : Knowing the non-zero vectors a and b, take any point A in the plane, make vector AB=vector a, and make vector AD=vector b when passing D point. According to the principle of vector, put the two vectors The first position of the translation is connected to form a single quadrilateral. Because DC is parallel to AB and AD is parallel to BC, according to the principle of parallelogram, the two sides of the quadrilateral are parallel to each other, and this quadrilateral is a parallelogram.
∵Connect AC to form a triangle ADC. According to the triangle rule of vectors: any two non-zero vectors, move the start point of one vector to the end point of the other vector, and the resultant force is from the start point of the first one to the end point of the second one.
∴Vector AC=vector a+vector b.
The triangle rule and parallelogram rule of vector addition reveal that any non-zero vector can be decomposed into any two directions of components, as long as the two components are translated and extended to form a graph that satisfies the parallelogram.
Any non-zero vector can be composed of array components, and is not limited to a fixed plane. Therefore, any non-zero vector has multiple transformation and composition possibilities, just like the many changes in the world, which seem to be different, but in reality Homologous.
Why can a multi-vector only be decomposed into two orthogonal decompositions?
Orthogonal decomposition refers to the method of decomposing a force into two mutually perpendicular components, Fx and Fy. Because of the definition of a vector, we know that as long as the length and direction of the vector remain unchanged, how the vector moves will be equal to the previous vector. When a vector is decomposed into two perpendicular vectors, the two mutually perpendicular vectors can actually be regarded as composed of multiple groups of small vectors that are perpendicular to each other in the plane, so it can be seen that each vector is composed of multiple groups of perpendicular to each other. Vector composition. From the perspective of vectors, as long as the size and direction are unchanged, the vectors are equal, so multi-vectors can only be decomposed into two orthogonal decompositions.
Any two non-collinear vectors can determine a unique plane. The vector can exist in one-dimensional space, two-dimensional space or three-dimensional space. The vector and scalar are different across multiple spatial dimensions, and the vector can move along itself It moves in parallel, so after determining a vector, the vector can theoretically exist anywhere, and the position is different, the vector is indeed the same.
With the cognitive ability of vectors, how ideas fly, and
the understanding of vectors, we have raised a dimension in thinking about classification. For example, a vector can be moved, and the vector after the movement is equal to the vector before the movement. Therefore, when we look at a thing, we can consider its characteristics and changes, whether it has changed the nature of the thing. If it is like a vector, look It seems to be different, but in fact it is the same. We also classify it into the same category as the things before the change. Doing so is similar to looking at the essence through the phenomenon, and the thinking of looking at things is deepened.
We regard the source of the vector as derived from the physical direction of gravity, which has both length and direction. Physics is closely related to our lives. The "calling" of vectors in physics is called "vectors". Vectors, vector files, etc. are often seen. Vectors contain independent separate images, which can be recombined freely and unlimitedly. , Its characteristic is that the image will not be distorted after zooming in. Compared with a single scalar, vector represents more complex and cognition and understanding. Knowing the concept of vector is like learning to reduce dimensionality and upgrade one-dimensional scalar thinking to three-dimensional vector thinking. Let us look at things more comprehensively and profoundly, and everything in the world moves the whole body. I used to look at things one-sidedly. Now that I have learned about vectors, I have begun to think in a deeper perspective and direction.
The learning of vector tells us how profound and superficial knowledge is in the field we don't know. The church respects science, sees its own shortcomings, and uses knowledge known to the public in different fields is also a kind of wisdom. There are no boundaries between disciplines, but our cognition creates obstacles to our thinking. In future studies, we will use all our knowledge and thoughts, without jumping to conclusions, arbitrarily qualitative things, standing on the shoulders of giants, Use universal axioms to create new theorems.