Russia 10 класс Geometry (advanced)
Chapters: 3
1. Lines and planes in space
Axioms · Parallelism and projection · Perpendicularity · Angles in space
- Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.
2. Polyhedra
Prism and pyramid · Symmetry
- Polyhedra: Prisms, Pyramids and the Platonic Solids – A polyhedron is a closed solid whose surface is made only of flat polygons (faces). Faces meet along edges, and edges meet at vertices. Prisms have two equal parallel bases joined by parallelograms; pyramids have one base and triangles meeting at an apex; a frustum is a pyramid with its top cut off by a plane parallel to the base. For every convex polyhedron, Euler's formula holds: V − E + F = 2. There are exactly five regular (Platonic) polyhedra.
3. Vectors in space
Vectors
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.