向量混合积与四面体体积公式的联系与应用  

The relationship and application of vector triple product and tetrahedron volume formula

作　　者：蔡婷 CAI Ting(Department of Mathematics and Statistics,School of Artificial Intelligence and Big Data,Hefei University,Hefei 230601,China)

机构地区：[1]合肥学院人工智能与大数据学院数学与统计系,安徽合肥230601

出　　处：《湖北师范大学学报（自然科学版）》2024年第1期23-28,共6页Journal of Hubei Normal University：Natural Science

基　　金：国家自然科学基金青年项目(12302034);合肥学院与湖北昶荣智能科技有限公司合作的横向项目(902/22050124013);合肥学院与安徽中科宇顺科技有限公司合作的横向项目(902/22050123131);合肥学院人才项目(21-22RC22);安徽省高校自然科学基金重点项目(No.KJ2021A0999)。

摘　　要：在解析几何中求解四面体的体积,传统的方法是选取一个顶点作为原点,并使用从这个顶点出发的三个不共面的向量,通过计算它们的混合积来得到四面体的体积。然而,实际上只需要从四个顶点中选择任意三个不共线的向量,计算它们的混合积即可求解四面体的体积。提出了一种更加通用的方法,利用向量混合积来求解四面体的体积,并且证明了此方法的准确性。最后,通过一个数值算例来说明这种方法的简便性和有效性。In analytical geometry,the traditional method to calculate the volume of a tetrahedron is to select one vertex as the origin and use three non-coplanar vectors originating from this vertex to obtainthe volume of the tetrahedron by calculating their scalar triple product.However,in reality,it is sufficient to choose any three non-collinear vectors from the four vertices and calculate their scalar triple product to determine the volume of the tetrahedron.This paper proposes a more general method that utilizes the vector triple product to calculate the volume of a tetrahedron and proves its accuracy.Finally,it adopts a numerical example to demonstrate the simplicity and effectiveness.

关 键 词：四面体体积 向量混合积 解析几何 

分 类 号：O182.2[理学—数学]