解析几何课程中求四面体体积新方法探究

A new method of computing the volume of a tetrahedron in analytic geometry course

对数学类专业开设的解析几何课程教材中求以不共面的4个点为顶点组成的四面体体积问题进行了研究。教材只给出了从四面体一个顶点出发的3个不共面向量求其混合积求体积的方法。事实上,只要从4个顶点中任取3个不共面向量,求其混合积就可以求四面体体积,并利用2种方法证明了所得结论。最后,以一个数值算例说明所用方法的正确性与有效性,对教材内容进行了深化与拓展。

This paper focuses on the problem of computing the volume of a tetrahedron in virtue of four vertexes not on the same plane in analytic geometry course for mathematics. The textbook only introduces that the volume of a tetrahedron is obtained by mixed product of three vectors which are emitted from one vertex and not on the same plane. However, the limited condition can be relaxed, that is, the volume of a tetrahedron can be obtained if the three vectors composed of four vertexes are not on the same plane, which is addressed in this paper. Two methods are adopted to prove the conclusion. Furthermore, a numerical example is presented to demonstrate the effectiveness of the proposed method and the results of textbook are deepended and widened in this paper.