History,applications,algebra and mathematical form of plane in mathematics (presentation.ppt)

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Presentation on PLANE Name: Nabeela Hayat Department: Bioinformatics Frontier Women University (Town Campus)

Table of contents ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Definition of Plane ,[object Object],[object Object]

A plane is a flat surface with no thickness. ,[object Object],[object Object],[object Object],[object Object],[object Object]

Plane Shape ,[object Object]

Examples ,[object Object],[object Object]

History ,[object Object],[object Object]

Planes in fiction ,[object Object]

Two-dimensional coordinate system ,[object Object]

Three-dimensional coordinate system ,[object Object],[object Object],[object Object],[object Object]

Orientation and handedness ,[object Object],[object Object],[object Object]

Area of Plane Shapes Triangle Area = ½b×h b = base h = vertical height Rectangle Area = b×h b = breadth h = height Trapezoid (US) Trapezium (UK) Area = ½(a+b)h h = vertical height Ellipse Area = πab Square Area = a2 a = length of side Parallelogram Area = b×h b = breadth h = height Circle Area = πr2 Circumference=2πr r = radius Sector Area = ½r2θ r = radius θ = angle in radians

Equations of plane ,[object Object],[object Object]

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

I) Equations of straight lines ,[object Object],[object Object],Rise Run P1(x1, y1) P1(x2, y2) y1-y2 x1-x2 x y l o

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Ф l x y Ф x y l o o Figure I a b Ф l x y o P2(x2, y2) Ф P1(x1, y1) P3(x2, y1) (x2-x1) (y2-y1) M N Case II Ф l x y o P2(x2, y2) Ф P1(x1, y1) P3(x2, y1) (x2-x1) (y2-y1) M N Case III Ф=0 l x y o Case I

I) Equations of straight lines ,[object Object],[object Object],[object Object],d>0 l x y o l x y o x y o d d d<0 d=0 (A) l ║ x-axis or l ┴ y-axis (B) l ║ x-axis or l ┴ y-axis (C) l ║ x-axis or l ┴ y-axis Figure1 (a) Figure1 (b) Figure1 (c)

I) Equations of straight lines ,[object Object],[object Object],[object Object],d>0 l x y o d (A) l ║ y-axis or l ┴ x-axis Figure2 (a) d<0 l x y o d (B) l ║ y-axis or l ┴ x-axis Figure2 (b) d=0 x y o (B) l ║ y-axis or l ┴ x-axis Figure2 (c)

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

iv) Derivation of Standard Forms of Equations of Straight Lines: ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Ф l x y o P(x1,y1) P'(x,y) Point-Slope Form Ф l x y o B(0,b) Slope Intercept Form A(a,0)

iv) Derivation of Standard Forms of Equations of Straight Lines: ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Ф l x y o P(x,y) Two-Points Form P1(x1,y1) P2(x2,y2) l x y o (0,b) (a, 0) P(x,y) Two Intercepts Form

iv) Derivation of Standard Forms of Equations of Straight Lines: ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Ф l x y o P(x,y) Normal Form B p A 90-Ф Q Ф P'(x,y) l x y o P(x1,y1) Symmetric Form

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

vi) To Transform the General Linear Equation in Standard Forms ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

vi) To Transform the General Linear Equation in Standard Forms ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],l O x y Figure 1(a) l O x y Figure 1(b) P(x1, y1) P1(x1, y') P(x1, y1)

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],l O x y Figure 1(a) P(x1, y1) l O x y Figure 1(b) P(x1, y1) P1(x1, y') Ф N M d Ф Ф

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],l O x y Figure 1 P(x1, y1) Q(x2, y2) d 1 l 2

I) Equations of straight lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],*Trapezoid: A four sided shape in which no sides are parallel l O x y Figure 1 P1(x1, y1) A(x1, 0) P2(x2, y2) P3(x3, y3) C(x1, 0) B(x1, 0)

II) Two and Three Straight Lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

II) Two and Three Straight Lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],l x y Figure 1 1 l 2 Φ Φ 2 1 θ O

II) Two and Three Straight Lines ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

II) Two and Three Straight Lines ,[object Object],[object Object],[object Object],[object Object],[object Object],F G E D A(x1, y1) C(x3, y3) B(x2, y2) Figure 1(a) F o E D A(x1, y1) C(x3, y3) B(x2, y2) Figure 1(b)

II) Two and Three Straight Lines ,[object Object],[object Object]

VI) Equation of straight lines in Matrix Form ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],x y x y

End of the presentation THANK YOU Give A BIG Hand

History,applications,algebra and mathematical form of plane in mathematics (presentation.ppt)

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