正交投影及其在几何上的应用

Orthogonal Projection and Its Application in Geometry

正交投影作为线性代数中非常重要的概念,它在许多具体问题中起着关键的作用.本文通过回顾向量在一维子空间和n维子空间上的正交投影,同时给出相应计算公式.作为应用,分别在二维平面和三维空间,建立了点到直线上和点到平面上的正交投影点、对称点和距离的计算公式.特别,利用所得计算公式,得到三维空间里点到直线的距离计算公式.进而,给出两个实例验证了相应的结论.

Orthogonal projection is a very important concept in linear algebra, which plays a key role in many specific problems. In this paper, we revisit the orthogonal projection of vector in one--dimensional subspace and n--dimension subspace. As application, formulas of the orthogonal projection point, the symmetry point and the distance, to straight lines or planes, are established in the two＇dimensional and three--dimensional spaces, respectively. Examples are given to verify the corresponding conclusion.