Some geometric properties in two and three dimensional generalized euclidean inner product spaces

In this thesis, which mainly consists of two chapters, it is aimed to determine some geometric properties in two and three-dimensional generalized Euclidean inner product spaces. In the first part, basic properties and reflection transformation in two and three-dimensional generalized Euclidean inner product spaces is obtained. The reflection transformation in these spaces is a transformation that takes a vector on an ellipse or ellipsoid to a vector on the same ellipse or ellipsoid. The matrix of the transformation is obtained first by using the generalized Euclidean inner product, then by using the generalized elliptic quaternions obtained by defining the generalized Euclidean vector product. Each method is proven and supported by examples. In the second part, the distance formulas between two basic geometric objects in these two and three-dimensional spaces are determined. The concept of distance here refers to the norm obtained from the generalized Euclidean inner product and the metric obtained from this norm. The distance formulas obtained are well known in analytic geometry; the distance formulas between a point and a line, a point and a plane, and two skew lines.