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Angles, areas, volumes: dot product and determinant
Ondič, Milan ; Beran, Filip (advisor) ; Zamboj, Michal (referee)
This bachelor thesis deals with the introduction of scalar product and determinant, which are important tools of analytic geometry. The purpose of the thesis is to provide a parallel interpretation of these two key concepts of advanced algebra - the dot product and the determinant - primarily from a geometric, not an algebraic, point of view. The aim of the thesis is to show how both representations can be derived just by solving geometric problems in two-dimensional space and then how to transfer them to three-dimensional space. The first part of the work is devoted to finding the angle between two vectors in the plane and to calculating the area of a triangle. Both of problems are solved in several ways and then the scalar product and determinant are derived. The second part of the work is devoted to three-dimensional space, in particular the angle between two vectors, lines and planes and the volume of a tetrahedron and parallelogram. This is then supplemented by the introduction of some notions of linear algebra, an investigation of the algebraic properties of the dot product and determinant, and a generalization of the notions to the n-dimensional space. The last part of the thesis is devoted to the analysis of selected czech high school mathematics textbooks in terms of the occurrence and...
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Dot product - definition and applications
Weissgráb, Lukáš ; Halas, Zdeněk (advisor) ; Rmoutil, Martin (referee)
This bachelor thesis presents various introductions of the dot product at several levels of difficulty. First part of the thesis deals with introduction of the dot product elementary, therefore only from knowledge of high school. The more advanced parts of this thesis are devoted to the introduction of the dot product as a bilinear form and focus on properties of this form. The final chapters are devoted to the Fourier series and the first fundamental form. All theorically explained pieces of knowledge are illustred with examples from mathematics and physics.
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Dot product in school mathematics
Krejčí, Veronika ; Halas, Zdeněk (advisor) ; Škorpilová, Martina (referee)
Title: Dot product in school mathematics Author: Veronika Krejčí Department: Department of Mathematics Education Supervisor: Mgr. Halas Zdeněk, DiS., Ph.D., Department of Mathematics Edu- cation Abstract: The theme of the bachelor thesis is the dot product in school mathe- matics. The bachelor thesis is adressed to the teachers, who teach at the secondary school. The first part consist of an analysis of the textbooks, the result of the ana- lysis is the reason to write this bachelor thesis and to build a new introduction to the theme of the dot product. In the second part we build the new introduction. The third and the fourth part consist of application of the dot product. Keywords: dot product, hyperplane, distance, angle 1
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