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. 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

Analytic Geometry of Space ( Analytic Geometry II )

Rubono Setiawan, M.Sc.

Mathematics Education

Sebelas Maret University


Assalamualaikum wr wb

Assalamualaikum Wr.Wb.


Course materials

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

Basic Competition 1 (KD 1)

  • 3 X Course in Class

  • 1 X Paper Based Test


Reference

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, 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

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

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 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

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

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

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

Distance of Two Points in Three Dimensional Space

  • Let two points in three dimensional space

    and

    the distance between the points and is


Problems

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 ?


Thank s

Thank’s


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