1. komponenta

Lecture type	Total
Lectures	30
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The goal of this course is to systematise, deepen and expand the students' notions of vectors and plane analytical geometry and to acquaint students with analytical geometry of the space.

COURSE DESCRIPTION AND SYLLABUS:
1. Axiomatics. Euclid's axioms. Cartesian model.
2. Vectors. Vectors as equivalence classes of oriented lines in plane and space. Magnitude, direction and orientation. Linear operations in set of vectors V2 and V3. Vector spaces V2 and V3. Colinearity and coplanarity. Linear independence of vectors. Basis for V2 and V3. Scalar product of vectors. Orthogonal projection of a vector onto the line and plane. Orthonormal basis. Vector product. Mixed product. (4 weeks).
3. Plane analytical geometry. Cartesian coordinate system. The distance between two points on a line and in the plane. Line in the plane. Ray and a half-plane. Point-line distance. The angle between two lines. Perpendicularity and parallelism. Circle. Polar coordinates in the plane. Transformation as a change in coordinate system. Conics. Directrix, focus and eccentricity. The equation of conics in Cartesian and polar system. The distance from focus. Conics in physics: the motion of planet and optical properties of conics. Transformation of coordinates (translation, rotation). Curves of the second order. Canonical form and clasiffication. Center, asymptotes and diameter. Some curves in plane (trochoid, Archimedean spiral). Linear operators and transformations. Isometries. (5 weeks).
4. Analytical geometry of space. Cartesian coordinate system. Plane and line in space. The distance between a plane and line. The angle between two planes. The angle between a line and plane. The angle between two lines. Common normal, distance between lines. Sphere. Cylindrical and spherical coordinate system. Transformation as a change in coordinate system. Linear operators and transformations. Isometries. Second-order surfaces. Classification (via symmetric operators). Some curves in space (spiral, Viviani's window). (5 weeks).

Mandatory course - Regular study - Mathematics