Transcript 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.
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 : 1. Rectangular Coordinates 2. Distance between two points 3. Orthogonal projection 4. Direction Cosines and Numbers of a line 5. Angle between two directed lines 6. Point dividing a segment in a given ratio 7. Polar Coordinates 8. Cylindrical Coordinates 9. 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 the coordinates are x, y, and z.
to 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
segment between them is bisected by the point if the
4/30/2020
Distance of Two Points in Three Dimensional Space
• Let two points in three dimensional space
P
2 (
x
2 ,
y
2 ,
z
2 ) and
P
1 (
x
1 ,
y
1 ,
z
1 ) is
P
2
P
1
P
2
P P
1
P
2 (
x
2
x
1 ) 2 (
y
2
y
1 ) 2
z
2
z
1 2 14
• • • •
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. 1.The xy-plane 2.The yz-plane 3.The xz-plane 4.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 ?