Analytic Geometry of Space ( Analytic Geometry II )

1. Analytic Geometry of Space ( Analytic Geometry II ) Rubono Setiawan, M.Sc. Mathematics Education Sebelas Maret University

2. Assalamualaikum Wr.Wb.

3. Course Materials 1. Coordinates : 3 – Dimensional Rectangular Coordinate Systems and Another Types of Space Coordinates 2. Planes and Lines 3. Sphere 4. Types of Surfaces 5. Quadric Surface

4. Basic Competition 1 (KD 1) • 3 X Course in Class • 1 X Paper Based Test

5. Reference • Snyder, V. and Sisam, C.H.,1914, Analytic Geometry of Space, Normood Press, J.S.Cusbing, Co.- Borwick & Smith Co., Massachusets, USA. • J.Stewart, Calculus, Fifth Edition, ITP, Singapore. • E.J. Purcell, Varberg, D., Rigdon, Calculus Ninth Edition, Prentice Hall, Inc, New-York, USA. etc

6. Tools, Software Tools 1. Ruler 2. Cross Section Paper Media : Geometrical Form in Space : Box, Tetrahedron, Prism, etc. Software : 1. Cabri 3D V2 2. Matematica

7. I. Coordinates Contents : • Rectangular Coordinates • Distance between two points • Orthogonal projection • Direction Cosines and Numbers of a line • Angle between two directed lines • Point dividing a segment in a given ratio • Polar Coordinates • Cylindrical Coordinates • Spherical Coordinates

8. 1. Rectangular Coordinates Coordinate Planes • Let them given three mutually perpendicular planes, XOY, YOZ, ZOX, intersecting at O, the origin. These planes will be called coordinate planes • The planes ZOX and XOY intersect in X’OX, the X-axis • The planes XOY and YOZ intersect in Y’OY, the Y-axis, the planes YOZ intersect in Y’OY, the Y-axis. • The planes YOZ and ZOX intersect in Z’OZ, the Z-axis

9. Distance Properties • Distance measured in the directions X’OX, Y’OY, Z’OZ, respectively, will be considered positive; those measured in the opposite directions will be regarded as negative • The coordinates of any point P are its distances from the three coordinate planes • The distance from the plane YOZ is denoted by x, the distance from the plane ZOX is denoted by y, and the distance from the plane XOY is denoted by z • These three number x, y, z are spoken of as the x, y, z – coordinates of P respectively

10. How to Determine Any Point in Rectangular Coordinates • Any Point P in space lies three real coordinates. Conversely, respectively, determine a point P, for if we lay off a distance OA = x cm the X-axis, OB = y the Y-axis, OC = z cm the Z – axis and then draw planes through A, B, C parallel to the coordinates are x, y, and z. • Lay off the distance OA = x on the X-axis. From A lay off the distance AD = y cm a parallel to the Y – axis. From D lay off the distance DP = z cm parallel to the Z-axis

12. Octantcs • 1.Figure - Octant.cg3 • The eight portions of space separated by the coordinate planes are called octans. • If the coordinates of a point P are a,b,c, the points in the remaining octants at the same absolute distances from the coordinate planes are (-a,b,c), (a,-b,c),(a,b,-c),(-a,-b,c),(-a,b,-c),(a,-b,-c),(-a,-b,-c)

13. Symmetric Properties • Two points are symmetric with regrad to a plane If the line joining them is perpendicular to the plane and the segment between them is bisected by the plane. • They are symmetric with regrad to a line, if the line joining them is perpendicular to the given line and the segment between them is bisected by the line • They are symmetric with regrad to a point if the segment between them is bisected by the point

14. Distance of Two Points in Three Dimensional Space • Let two points in three dimensional space and the distance between the points and is

15. Problems • Plot the following point to scale, using cross section paper : (1,1,1), (2,0,3), (-4,1,5), (0,0,-7), (7,6,4), (-3,-2,-5) • Given point (k, l, m), write the coordinate of the point symmetric with it as to the plane XOY; the plane ZOX ; the X-axis ; the Y-axis ; the origin. • Find the distance from (5,-4,-3) to each of the following. • The xy-plane • The yz-plane • The xz-plane • The x- axis • Which of the points P (6,2,3), Q(-5,-1,4) and R(0,3,8) is closest to the xz- plane? Which point lies in the yz-plane ?

16. Thank’s