# Analytic Geometry of Space ( Analytic Geometry II ) - PowerPoint PPT Presentation

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Analytic Geometry of Space ( Analytic Geometry II ). Rubono Setiawan, M.Sc. Mathematics Education Sebelas Maret University. Assalamualaikum Wr.Wb. Course Materials. 1. Coordinates : 3 – Dimensional Rectangular Coordinate Systems and Another Types of Space Coordinates

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Analytic Geometry of Space ( Analytic Geometry II )

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## Analytic Geometry of Space ( Analytic Geometry II )

Rubono Setiawan, M.Sc.

Mathematics Education

Sebelas Maret University

### 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

### Basic Competition 1 (KD 1)

• 3 X Course in Class

• 1 X Paper Based Test

### 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

### 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

### 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

### 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

### 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

### 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

### 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)

### 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

### Distance of Two Points in Three Dimensional Space

• Let two points in three dimensional space

and

the distance between the points and is

### 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 ?