摘要：向量是现代数学中重要和基本的数学概念之一，它是沟通代数、几何与三角函数的一种工具，有着极其丰富的应用背景．因为向量具有良好的运算通性、几何的直观性、表述的简洁性和处理问题的一般性，所以向量法有广泛的应用，对各种数学问题的融会贯通、几何证明能力的提高都有一定的帮助．
但调查表明受传统数学知识的负迁移，无论是对向量的基础知识的掌握，还是向量运用的推广，甚至对向量教学的探讨都是今后一个艰难的课题．本文尝试在这三个方面作一些较深入的整理和研究．以加深对向量的工具性作用的认识，推广向量法．
本文从高中引入向量的必要性出发，先介绍了向量的基础知识，对那些暂时还没有被选入现行中学教材中的运算和向量空间等基础知识也作了归纳，以完善向量知识体系，然后对编入教材中的向量内容进行分析，通过例题加注解的形式对向量在高中数学中的应用进行了详细的分类阐述，并且与传统方法作了对比性的评注，这是本文的重点之一，本文的另一个重点是对高中向量的教与学的现状进行了调查与分析.
主要结论之一：师生都认为向量有用、可用，可就是用的不踏实，向量法在高中数学教学方法中还只是个“候补队员”，通常是传统方法失效后才想起向量法，远远没有达到普及的程度.主要结论之二：用传统方法解某些立体几何题所用的时间与用向量方法所用时间尽管没有显著差异，但是用向量方法在解题的准确性上明显优于传统方法，而且所用的时间也略少于传统方法．最后是对向量教学的几点思考，期望能给今后的向量教学研究以借鉴．
关键词：向量；向量的应用；向量法；中学教学
Abstract ：Vector is one of the important and basic concepts in modern mathematics. It is also a tool that links up algebra， geometry， and trigonometric. It has widely applied in solving problems for its general operations， geometrical visualization， and conciseness in expressing ideas. Therefore， the teaching and learning of vectors does not help students build an integrated body of mathematical knowledge， but also enhance their abilities to solve the proof problems in geometry.
Relative literature shows that it is still a very difficult topic in school mathematics due to the negative influence of traditional mathematic knowledge. Specifically， both teachers and students experience a lot of difficulties in grasping the basic knowledge of vector， in promoting the application of vector， and in the teaching and learning of vectors.
In this thesis， we shall first focus on the basic knowledge about vectors including those that are already presented in the current mathematics textbooks used in schools and those that are not yet presented in school textbooks， but basic in certain sense. This effort is made to form a complete structure of basics in vector.
We then make a systematic analysis of those topics in vector in school mathematics. Particularly， we discuss the various applications of vector in school mathematics with examples. We also compare its applications with those traditional methods. This kind of comparison helps us see the advantages of vector method over traditional methods.
Finally we did a survey to investigate the current status of vector’s teaching and learning in schools. We found that both teachers and students are aware of the potential usage of vectors. However， they only use it when the traditional methods do not help them solve their problems. In this case， the vector method is used as a backup. We also found that the vector method obviously superior to the traditional methods in the accuracy of mathematic problem solving and the vector method is a little more efficient than the traditional methods even though the difference has not reached a significance level in solving solid geometrical problems.
Some suggestions are also made at the end of the thesis for improving the teaching and learning of vectors in schools.
Keywords: vector; application of vectors， vector method， teaching and learning
引 言
数学的价值在于它的广泛应用性．向量集数形于一身，沟通了代数、儿何、三角，向量可以由其几何直观来确定方位、研究形状、约简推理等．向量是代数的研究对象，是一种运算．它的运算对象从数到字母到代数式再到向量，运算也从一元到多元；向量又是几何的研究对象，向量有方向、有长度，能建立直线和平面的向量结构．引入向量，建立空间的向量结构为解析几何奠定了理论基础，为使用代数方法研究几何找到了更强有力的工具．向量在现代物理学领域也有广泛的应用．
将向量作为高中数学的必学内容，是必然的．无论是从国内外中学数学教学改革的历史经验来看，还是从当前中学数学教学的目的来看，向量进入中学数学，对于学生更好地学习几何、代数，将来进一步学习高等数学，对于学生灵活运用数学知识解决实际问题都会有启蒙和奠基的作用．
尽管向量是数学的重要概念之一，但最初并未被数学家们所接受，人们对向量的认识有一个过程．向量内容也很丰富，除了向量的起源和发展、基本概念、性质、运算等与中学数学联系比较密切的相关基础知识，还有向量积、混合积、n维向量空间等知识．学习向量有助于学生弄清数学的各个分支知识之间的联系，掌握中学数学思想和数学方法，更深地认识到数学研究的是现实世界的数量关系和空间形式的统一．
