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Analytic Geometry of Space ( Analytic Geometry II )PowerPoint Presentation

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

5. Quadric Surface

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 ?

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