基于矩阵理论的高考立体几何法向量求解与教学启示

Solution of Normal Vectors in Solid Geometry for College Entrance Examination Based on Matrix Theory and Teaching Implications

摘要: 《普通高中数学课程标准(2017年版2020年修订)》强调“立体几何与空间向量”对培养学生核心素养的重要性。针对学生备考中机械套用向量公式求解法向量、缺乏本质理解的问题，文章对比传统坐标法与矩阵理论法，发现传统方法依赖特殊坐标系且计算繁琐。引入矩阵理论，通过二阶、三阶行列式及向量叉乘解析法向量求解逻辑，以多道高考题验证其适用性，总结“建系–定向量–算法向量–应用”的解题步骤。教学启示部分提出，教师需融合高等数学原理构建完整思维链，结合矩阵运算与几何直观突破公式记忆瓶颈，从高阶视角培养学生抽象建模能力，落实核心素养。

Abstract: The “General Senior High School Mathematics Curriculum Standards (2017 Edition, Revised in 2020)” emphasizes the significance of “solid geometry and spatial vectors” in cultivating students’ core literacy. Aiming at the problem that students mechanically apply vector formulas to solve normal vectors during exam preparation and lack essential understanding, this article compares the traditional coordinate method with the matrix theory method and finds that the traditional method relies on special coordinate systems and is computationally cumbersome. The matrix theory is introduced. The solution logic of normal vectors is analyzed through second-order and third-order determinants and vector cross products. Its applicability is verified by multiple college entrance examination questions. The solution steps of “establishing the system—determining the vector—algorithm vector—application” are summarized. The teaching enlightenment section proposes that teachers need to integrate the principles of advanced mathematics to construct a complete thinking chain, combine matrix operations and geometric intuition to break through the bottleneck of formula memory, cultivate students’ abstract modeling ability from a higher-order perspective, and implement core literacy.