A Collection of Problems in Analytical Geometry

A Collection of Problems in Analytical Geometry, Part II: Three-Dimensional Analytical Geometry is a collection of problems dealing with analytical geometry in the field of theoretical mechanics. The book discusses rectangular Cartesian coordinates in three-dimensional space and the division of an interval in a given ratio. The sample questions concern problems dealing with isosceles triangles, vertices, and center of gravity of equal masses.
The book defines the concept of a vector and then lists problems concerning the triangle law and the scalar product of two vectors. Other problems focus on the equations of a surface and a curve and on questions related to the intersection of three surfaces. The text lists other problems such as the equation of a plane, the direction-vector of a straight line, and miscellaneous problems pertaining to the equations of a plane, of a straight line, and of a sphere in a direction-vector.
The selection is useful for professors in analytical geometry and for other courses in physic-mathematics and general engineering.

Inhalt

Author's Preface to the First Russian Edition
Author's Preface to the Second Russian Edition
Part II. Three-dimensional Analytical Geometry

6. Elementary Problems of Three-dimensional Analytical Geometry
   § 27. Rectangular Cartesian Coordinates in Three-Dimensional Space
   § 28. Distance between Two Points. Division of an Interval in a Given Ratio
7. Vector Algebra
   § 29. The Concept of a Vector. Projection of a Vector
   § 30. Linear Operations on Vectors
   § 31. The Scalar Product of Two Vectors
   § 32. The Vector Product of Two Vectors
   § 33. Scalar Triple Product
   § 34. Vector Triple Product
8. Equation of a Surface and Equations of Curves
   § 35. Equation of a Surface
   § 36. Equations of a Curve. The Problem of the Intersection of Three Surfaces
   § 37. Equation of a Cylindrical Surface with Generators Parallel to One of the Coordinate Axes
9. Equation of a Plane. Equations of a Straight Line. Equations of Surfaces of the Second Order
   § 38. The General Equation of a Plane. Equation of a Plane Passing through a Given Point and Having a Given Normal Vector
   § 39. Degenerate Equations of a Plane. Equation of a Plane in Terms of Its Intercepts on the Coordinate Axes
   § 40. The Normal Equation of a Plane. Distance of a Point from a Plane
   § 41. Equations of a Straight Line
   § 42. The Direction-Vector of a Straight Line. The Canonical Equations of a Straight Line. Parametric Equations of a Straight Line
   § 43. Miscellaneous Problems Relating to the Equation of a Plane and the Equations of a Straight Line
   § 44. The Sphere
   § 45. Equations of a Plane, a Straight Line, and a Sphere in Vector Notation
   § 46. Surfaces of the Second Order
   Appendix: Elements of the Theory of Determinants
   § 1. Determinants of the Second Order: Systems of Two Linear Equations in Two Unknowns
   § 2. Homogeneous System of Two First-degree Equations in Three Unknowns
   § 3. Determinants of the Third Order
   § 4. Properties of Determinants
   § 5. Investigation and Solution of a System of Three Linear Equations in Three Unknowns
   § 6. Determinants of the Fourth Order
   Answers to Questions and Notes on Solutions