Multiplication of vectors (what is the meaning of learning vectors?)

Since the start of the new curriculum reform, the teaching of teachers and the learning of students have undergone tremendous changes. This change is affected by the new teaching model on the one hand, and by changes in the content of teaching materials on the other hand. For example, two notable changes in high school mathematics textbooks are the introduction of “vectors and derivatives”. The purpose of introducing these two pieces of knowledge is to provide new means for studying the graph between functions and spaces.

Derivative is a knowledge content that many people are very familiar with, and it has now become an important and popular test site for mathematics in the college entrance examination. For the cognition of vector, many people only stay at the “tool” level, and do not fully realize the importance of vector thinking.

The introduction of vector-related knowledge has certain practical significance for our high school mathematics education. For example, vectors in space have more advantages than traditional knowledge and methods in solving solid geometry. In mathematics learning, use the coordinate method of vectors between spaces to solve the “three corners” problem between spaces. We found that this method is better and more operable than the traditional solution, because as long as the system can be established, there are coordinates.

Although we recognize the position of vector in high school mathematics education and realize that the knowledge content related to vector has a very important position and educational value in mathematics education, many people do not understand vector related knowledge in practical applications. The knowledge conclusions do not have a deep understanding, and some students just rely on rote memorization to digest the vector knowledge content, which is completely contrary to the spirit of the new curriculum reform.

The instrumental characteristics of vectors are reflected in many branches of mathematics, especially in advanced mathematics and analytic geometry, where the idea of vectors permeates widely. In high school mathematics learning, vector is one of the compulsory courses, which can cultivate students’ mathematical ability and literacy, and help students improve their comprehensive mathematical ability.

What is a vector? Where do vectors come from?

We know that in physics, a quantity that has magnitude but no direction is called a scalar, and a physical quantity that has both direction and magnitude is called a vector. Vectors are widely used in high school physics learning, such as force, velocity, acceleration, electric field strength, etc. In fact, the vector in physics is the vector in mathematics, but the same quantity has two different names in different disciplines.

In physics and engineering, geometric vectors are often called vectors. Many physical quantities are vectors, such as the displacement of a physical resource ***, the force on a ball hitting a wall, and so on. On the contrary, it is a scalar, that is, a quantity with only magnitude but no direction. Some vector-related definitions are also closely related to physical concepts, such as the vector potential corresponding to potential energy in physics.

Around 350 BC, the famous ancient Greek scholar Aristotle knew that force can be expressed as a vector, and the resultant force of two forces can be obtained through the famous parallelogram law.

One of the British scientists, Newton, used directed line segments to represent vectors. The word “vector” comes from directed line segments in mechanics and analytic geometry.

As we all know, in mathematics, we use magnitude and direction vectors to call quantities. Meanwhile, vectors are also called Euclidean vectors, geometric vectors, and vectors.

A vector can be represented visually as a line segment with an arrow. The arrow represents the direction of the vector, and the length of the line segment represents the magnitude of the vector.

Vectors only correspond to size, and quantities without direction are called quantities, which are called scalars in physics.

Vectors were first used in physics, such as force, velocity, displacement, electric field strength, magnetic induction and many other physical quantities are vectors. This also reflects the “intimate relationship” between mathematics and physics, and the importance of mathematics as a basic subject.

How to represent a vector?

Generally bold letters are written in print, such as A, B, U, V, etc. Also add a small arrow “→” at the top of the letter.

If you are given a starting point (A) and an ending point (B) of a vector, you can denote the vector as AB with a → sign at the top of the letter.

In the Cartesian coordinate system between spaces, vectors can also be expressed in pairs, such as (2, 3) is a vector in the Oxy plane.

Related definitions of vectors include sliding vectors, fixed vectors, position vectors, direction vectors, opposite vectors, parallel vectors, co-planar vectors, normal vectors, etc. Vectors are generally defined as elements between vectors spaces. It should be noted that these abstract vectors are not necessarily represented by pairs of numbers, and the concepts of magnitude and direction do not necessarily apply. For example, the concept of geometric vectors is abstracted in linear algebra to obtain more general vector concepts.

So in the process of learning mathematics, you must strengthen the study of basic knowledge and further understanding, so that you can learn to distinguish what kind of concept “vector” is based on the context.

As long as we have a good grasp of the relevant knowledge, we can set the coordinate system according to a basis between vector spaces, and define the norm and Inner product, which allows us to compare abstract vectors with concrete geometric vectors.

Seeing the expression of vectors, we can easily think of the mathematical knowledge of complex numbers. In fact, important knowledge content such as vector has entered the field of mathematics and achieved important development, thanks to the development of complex related knowledge content.

It will take hundreds of years before and after complex numbers to establish a complete knowledge system. But the vector structure of spaces in the history of mathematics has been known to mathematicians for quite a long time. Until the end of the 18th century, the Norwegian surveyor Wiesel first used points on the coordinate plane to represent complex numbers a+bi, and used geometrically meaningful complex number operations to define vector operations.

People use vectors to represent points on the coordinate plane, and use the geometric representation of vectors to study geometric problems and trigonometry problems.

During the development of complex numbers, mathematicians found that their use was sometimes limited. If there are forces acting on the same object that are not in the same planeIn the above, it is necessary to find the so-called three-dimensional “complex numbers” and the corresponding operation system.

In the middle of the 19th century, the British mathematician Hamilton invented the quaternion, which includes a quantity part and a vector part to represent a vector between spaces. Thereafter, Hamilton laid the foundations for the establishment of vector algebra and vector analysis.

British mathematician and physicist Maxwell separated the quantity part of the quaternion from the vector part, thus creating a large number of vector analysis.

In the 1880s, Gilbert and Hayside in the UK independently completed the creation of three-dimensional vector analysis and the splitting of quaternions.

They propose that a vector is just the vector part of a quaternion, but it is not independent of any quaternion. They introduced two types of multiplication, the scalar product and the cross product. Generalizes vector algebra to vector calculus with variable vectors.

Therefore, when the knowledge of complex numbers is gradually accepted and used for further research in mathematics, it also directly promotes mathematicians to use complex numbers in the planar yo-yo resource*** Represent and study vectors, and connect the properties of space with vector operations, making vectors a mathematical system with excellent versatility in operations.

Introduce vector-related knowledge into high school mathematics education, allowing students to study and research vectors in a systematic and in-depth manner. The purpose of this is not only to learn vector knowledge, but also to help our students better understand vector-related knowledge in physics classes. At the same time, students can better understand vectors by learning vector content in physics. For example, in mechanics, the decomposition and synthesis of forces, velocities, etc. Apply the theory of addition and subtraction of vectors.

So we must take vector learning seriously and lay a solid foundation for future learning. In the usual process of learning mathematics, one must first master the basic knowledge of the vector method, learn to master and use the thinking method of the vector, learn to reasonably reorganize and integrate the mathematical knowledge and mathematical thinking methods of each part, and use the vector to use the connection, movement, From an aesthetic point of view, make vertical and horizontal connections and extensive associations.

We often say that mathematics comes from life, and at the same time it must be able to serve life. Problems in life should be solved by converting them into specific mathematical problems with mathematical methods, such as equations, vectors and so on. The practical application of vector-related knowledge can not only reflect its instrumentality, but also fully reflect the teaching value of vector in improving students’ mathematical ability.