3.2 Affine Coordinate System and Rectangular Coordinates System

1. 3.2 Affine Coordinate System and Rectangular Coordinates System Rene Descartes

2. I.Affine coordinate system Lemma 3.1Fetch a vector that not equal to zero in a line, then for every vector in the line, there exists an unique fixed real number , such that Lemma 3.2Fetch two non-collinearvectors in a plane, then for every vector in the plane, there exists an unique binary ordered real number such that

3. Theorem 3.3Fetch three non-coplanar vectors in space, then for every vector in the space, there exists an unique ternary ordered real array such that Definition 3.11Fetch a point in space and three ordered non-coplanar vectors such that they con- struct an affine ordinate system , then the point is called coordinate origin, are called coordinate vector or basis vector, basis for short, the line they lie in are

4. called axis, axis, axis, respectively, coordinate axis for all. Definition 3.12For vector in space, it’s decomposition expression is denoted by under affine coordinate system , is called coordinate of the vector under coordinate system Definition 3.13For a point , vector is called vectorradial of under affine coordinate system ,

5. the coordinates of vector radial are called affine coordinate ofunder this coordinate system. • Remark As three coordinate vectors may have • two unlike position relationship presented by graphs below. • right hand affine coordinate system; • (b) left hand affine coordinate system. (b) (a)

6. Example 3.4In the graph trapezoid and is the midpoint of , find coordinates of the point and vector under coordinate system . II. Vector operations by coordinate Let , then the additive (subtractive) operation of vectors are as follows.

7. or Scalar product. or Example 3.5Let compute

8. Example 3.6For given two points find coordinates of the vector radials and • Conditions for collinear and • coplanar of vectors Theorem 3.4 The necessity and sufficient condition of coplanar for three vectors

9. is that Example 3.7 Let three vectors are coplanar, compute

10. IV. Coordinate of scaling point of division Let be two arbitrary points in space,then, the coordinate of point that divides line segment into scalar is , where Specially,

11. Example 3.8Let find the coordinate of barycenter of Answer

12. V. Rectangular coordinates system in space Rectangular coordinates system is a kind of special affine coordinate system. 1. Rectangular Coordinates system in Space Rectangular coordinates system in space is constructed by three number axes, which via a fixed pointare vertical each other, following the right hand law.

13. Ⅱ Ⅲ Ⅳ Ⅰ Ⅵ Ⅶ Ⅷ Ⅴ z axis • coordinate origin • coordinate axis plane plane • coordinate plane Zox plane • 8 octants y axis X axis

14. 2. Distance formula between two points For we have distance formula between two points

15. 3. Direction angle of vector In rectangular coordinates system in space, angles of vector and three coordinate vectors are called direction angle of , cosine of the direction angles are called direction cosine of . Let then

16. Property of direction cosine.

17. Example 3.9 Find a point in axis such that it has the equal distance to the following two points Example 3.10 Given two points compute the direction cosine of vector

18. The End of Section 3.2