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

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Presentation on PLANE

Name: Nabeela HayatDepartment: Bioinformatics

Frontier Women University (Town Campus)

Table of contentsDefinition of Plane

Plane Shape ExamplesHistory

Planes in fiction

Applications

Two-dimensional coordinate system Three-dimensional coordinate systemOrientation and handednessArea of Plane ShapesEquations of plane

Definition of Plane

A plane is a flat surface with no

thickness.It extends

forever.

A plane is a flat surfacewith no thickness.

Our world has three dimensions, but

there are only two dimensions on a

plane .

Examples:

length and height, or

x and y

And it goes on forever .

Plane Shape A 2 dimensional shape. Has width and breadth,

but no thickness.

These are plane shapes.

Examples

When we draw something on a flat piece of paper we are drawing on a plane ...... except that the paper itself is not a plane, because it has thickness! And it should extend forever, too.So the very top of a perfect piece of paper that goes on forever is the right idea

History Cartesian means relating to the French

mathematician and philosopher René Descartes (Latin: Cartesians), who, among other things, worked to merge algebra and Euclidean geometry. This work was influential in the development of analytic geometry, calculus, and cartography.

The idea of this system was developed in 1637 in two writings by Descartes and independently by Pierre de Fermat, although Fermat did not publish the discovery.[1] In part two of his Discourse on Method, Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides. [citation needed] In La Geometries, he further explores the above-mentioned concepts.

Planes in fiction

The 1884 novel Flatland by Edwin A. Abbott features the concept of a geometric, two dimensional infinite plane inhabited by living geometric figures (triangles, squares, circles, etc.). It has been described by Isaac Asimov, in his foreword to the Signet Classics 1984 edition, as "the best introduction one can find into the manner of perceiving dimensions".

Applications Cartesian coordinates are often used to represent two or

three dimensions of space, but they can also be used to represent many other quantities (such as mass, time, force, etc.). In such cases the coordinate axes will typically be labeled with other letters (such as m, t, F, etc.) in place of x, y, and z. Each axis may also have different units of measurement associated with it (such as kilograms, seconds, pounds, etc.). It is also possible to define coordinate systems with more than three dimensions to represent relationships between more than three quantities. Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces, which can become rather complicated.) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three (perhaps two or three of many) non-spatial variables.

Two-dimensional coordinate system

A Cartesian coordinate system in two dimensions is commonly defined by two axes, at right angles to each other, forming a plane (an xy-plane). The horizontal axis is normally labeled x, and the vertical axis is normally labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is added, providing a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles, and such systems are occasionally used today, although mostly as theoretical exercises.) All the points in a Cartesian coordinate system taken together form a so-called Cartesian plane. Equations that use the Cartesian coordinate system are called Cartesian equations.

Three-dimensional coordinate system

The coordinate surfaces of the Cartesian coordinates (x, y, z).The three dimensional Cartesian coordinate system provides the three physical dimensions of space length, width, and height .

The xy-, yz-, and xz-planes divide the three-dimensional space into eight subdivisions known as octants, similar to the quadrants of 2D space. While conventions have been established for the labeling of the four quadrants of the x-y plane, only the first octant of three dimensional space is labeled. It contains all of the points whose x, y, and z coordinates are positive.The z-coordinate is also called applicate.

Orientation and handedness

The right hand rule as an perfect example Fixing or choosing the x-axis determines the y-

axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane.

The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the "first" and the y-axis the "second" axis) is considered the positive or standard orientation, also called the right-handed orientation.

Area of Plane Shapes

TriangleArea = ½b×hb = baseh = vertical height

RectangleArea = b×hb = breadth

h = height Trapezoid (US)Trapezium (UK)Area = ½(a+b)hh = vertical height

EllipseArea = πab

SquareArea = a2

a = length of side

ParallelogramArea = b×hb = breadthh = height

CircleArea = πr2 Circumference=2πrr = radius

SectorArea = ½r2θ r = radius

θ = angle in radians

Equations of plane

I. Equations of straight lines

II. Two and Three Straight Lines

I) Equations of straight linesi) The slope of a Lineii) The slope of a Line Joining Two Pointsiii) Equations of a straight line Parallel and perpendicular to

coordinate axesiv) Derivation of Standard Forms of Equations of Straight Linesv) A linear Equation in Two variables represents a Straight Linevi) To Transform the General Linear Equation in Standard Formsvii) Position Of a Point with respect to a Line:viii) Distance of a point from a Line:ix) Distance Between Two Parallel Lines:x) Area of a Triangular Region Whose Vertices are Given:

I) Equations of straight lines

i) The slope of a LineLet l

be any non vertical line in the plane. Let P1(x1, y1) and P2(x2, y2) be any distinct points on l as shown in figure. If we move along l from P1 to P2 we move y1- y2 units in the y-direction and x1-x2 units in the x-direction.

I) Equations of straight lines ii) The slope of a Line Joining Two

PointsThe angle of inclination of

a line l is the smallest angle between angle Ф measured counter clock-wise from the direction of positive x-axis to l (Refers to figure 1)

a) Case I When Ф =0°:If Ф=0°, the line is horizontal and so the slope m=0 since Ф =0° so tan Ф =tan0°=0=m (Refers to figure Case I)

b) Case II When 0°< Ф <90°Draw perpendiculars P1M & P2N on x-axis & perpendicular P1P3 on P2N. Then angle P2P1P3= Ф, P1P3=x2-x1.In triangle P1P3P2, we have m=y2-y1/x2-x1=tan Ф (Refers to figure Case II)

c) Case III When 90°< Ф <180° tan(π- Ф)= y2-y1/x1-x2

- tan Ф= y2-y1/x1-x2

(Refers to figure Case III)


iii) Equations of a straight line Parallel and perpendicular to coordinate axes:

a) Equations of a straight line Parallel to x-axis or perpendicular to y-axis:Every point on l has its distance from x-axis equal to d which represents the y-coordinate. Thus, all the points on l satisfy the equation y=d known as Equations of a straight line Parallel to x-axis. (Refers to figure 1(a), (b), (c)


iii) Equations of a straight line Parallel and perpendicular to coordinate axes:

b) Equations of a straight line Parallel to y-axis or perpendicular to x-axis:

Every point on l has its distance from y-axis equal to d which represents the x-coordinate. Thus, all the points on l satisfy the equation x=d known as Equations of a straight line Parallel to y-axis. (Refers to figure 2(a), (b), (c)


iv) Derivation of Standard Forms of Equations of Straight Lines:

There are six standard forms of equations ofa) Slope Intercept Formb) Point-Slope Formc) Two-Points Formd) Two Intercepts Forme) Normal Formf) Symmetric Form

iv) Derivation of Standard Forms of Equations of

Straight Lines:

a) Slope Intercept Formif a line l intersects the x-axis at a point A (a, 0) then the number a is called the x-intercept of the line l. if a line l intersects the y-axis at a point B(0, b) then the number b is called the y-intercept of the line l

b) Point-Slope FormTheorem: An equation of

non –vertical line l with slope m that passes through the point P(x1 , y1) is given by

y-y1=m(x-x1)


Straight Lines:c) Two-Points Form

Theorem: An equation of non-vertical line l passing through two points P1 (x1 , y1) and P2(x2, y2) is given by

y-y1=(y2-y1)(x-x1)/(x2-x1)This is called Two-Points

Form of a lined) Two Intercepts Form

Theorem: An equation of a line l whose non-zero x & y intercepts are a & b respectively, is given byx / a +y / b=1

This is called Two-intercepts Form of a line


Straight Lines:e) Normal Form

Theorem: An equation of non-vertical line l, when P is the length of the perpendicular from the origin to the line l and Ф is the inclination of the perpendicular, is given by

x cos Ф + y sin Ф = P f) Symmetric Form

Theorem: An equation of non-vertical line l with inclination Ф passing through a point P(x1 , y1) is given by

x-x1/cosФ = y-y1/sinФ=rDirected distance from P(x1, y1)

to any point P’(x, y) on lThis is called symmetric form

of the line

I) Equations of straight linesv) A linear Equation in Two variables represents a Straight Line

Every linear equation of two variables is given byax + by +c = 0 --------- (1)

a) Case IIf a ≠ 0, b = 0

Then equation 1 becomesax + c = 0 x = -c/aWhich is the equation of a straight line parallel to y-axis at a directed distance –c/a from y-axis

b) Case IIIf a = 0, b ≠ 0

Then equation 1 becomesby + c = 0 y=-c/bWhich is the equation of a straight line parallel to x-axis at a directed distance –c/b from x-axis

c) Case IIIIf a ≠ 0, b ≠ 0

Then equation 1 becomesax + by + c = 0b) by = -ax -cc) y = - ax/b - c/bWhich is the slope-intercept form of a straight line with slope m = -a/b & k = -c/b


vi) To Transform the General Linear Equation in Standard Forms:

a) Slope Intercept Formb) Point-Slope Formc) Two-Points Formd) Two-Intercepts Forme) Normal Formf) Symmetric Form

vi) To Transform the General Linear Equation in Standard Forms

ax + by + c = 0 -------------- (1)

a) Slope Intercept Form:Equation 1 can be written asy= – ax/b – c/b = mx + kwhere slope m = –a/b & k= – c/b which is the required transformed slope intercept form of equation 1

b) Point-Slope Form:The slope of a line ax + by + c = 0 is m = –a/bA point on the line is (- c/a, 0)Therefore the transformed equation isy – 0 = – ax/b – [ x – ( – c/a)] y = – ax/b – ( x + c/a)

vi) To Transform the General Linear Equation in Standard

Formsc) Two-Points Form:

To transform the line ax + by + c = 0 to two points-form, we chose two arbitrary points on the line (- c/a, 0) & (0, - c/b) are two such points on the line.

The equation of the line through these points is y – 0 = (- c/b – 0) / (0 + c/a)[x – (- c/a)]

y = – a/b ( x + c/a)

d) Two-Intercepts Form:We have ax + by + c = 0 ax + by = –c ax/ -c + by/ -c = 1 (Dividing both sides by –

c) x/ - c + y/ - c = 1 (where – c/a & -c/b are

respectively the x-intercept & y-intercept)

Which is the required Two-Intercepts Form of the line

vi) To Transform the General Linear Equation in Standard Forms

e) Normal Form:We know that an equation of a line in a normal form isx cos Ф + y sin Ф = pLet r cos Ф = a & r sin Ф =bThen

r2 cos2 Ф =a2 & r2 sin2 Ф =b2

so r2 cos2 Ф + r2 sin2 Ф = a2+ b2

r2 ( cos2 Ф + sin2 Ф ) = a2+ b2

r2 = a2+ b2

r = ± √a2+ b2

f) Symmetric Form:We have sin Ф = b/ ± √a2+ b2 , cos Ф = a/ ± √a2+ b2

a point on ax + by + c = 0 is ( - c/a, 0)Thus the equation in the symmetric form becomes

[{ x – ( - c/a )} / a] / ± √a2+ b2 = (y – 0 /b) / ± √a2+ b2 = rWhich is the required transformed equation. Sign of radical is to

be properly choosen.

I) Equations of straight linesvii) Position Of a Point with respect to a Line:

(Refers to figure 1 (a)Theorem: Let P(x1 , y1) be any point in plane &

l: ax + by + c = 0 --------------------- (1)Be a given line in the plane not containing the point P.Then P lies(a) Above the line l if

ax1 + by1 + c > 0(b) Below the line l if

ax2 + by2 + c < 0(Refers to figure 1 (b)

1) Corollary IThe point P(x1, y1) is above or below the line l: ax + by + c = 0 if ax1 + by1 + c & b have the same sign or have opposite sign respectively

Corollary IIThe point P(x1, y1) and the origin O(0, 0) are: On the same side of l if ax1 + by1 + c & c have the same sign On the opposite side of l if ax1 + by1 + c & c have the opposite sign


viii) Distance of a point from a Line:Theorem: The distance d from the point P(x1,

y1) to the line

l: ax + by +c = 0 is given byd = │ax1 + by1 + c│/ √a2 + b2

a) Corollary IIf l is horizontal, thend = │by1 + c /a│

Corollary IIIf l is vertical, thend = │ax1 + c /a│


xi) Distance Between Two Parallel Lines:Let l1 & l2 be two parallel lines & P(x1,

y1) be & y point on l1. Draw a perpendicular from P on l2 which meets it at the point Q(x2, y2)

The distance from l1 to l2

(or from l2 to l1) is the length of the perpendicular PQ i.e. d = │ PQ│

In general the distance between two parallel lines is the distance from any point on one of the line to the other line.


x) Area of a Triangular Region Whose Vertices are Given:Let ABS be the triangle whose vertices are P1(x1 , y1),P2(x2, y2) & P3(x3, y3). Draw perpendicular P1A, P2C & P3B on x-axisArea of the triangular region P1P2P3

= Area of the trapezoidal* region P1ABP3 + Area of the trapezoidal region P3BCP2 - Area of the trapezoidal region P1ACP2

= [(│P1A │+ │P3B │)(│AB│)]/2 + [(│P3B │+│P2C │)(│BC│)] – [(│P1A │+│P2C │)(│AC│)]/2= [(y1 + y3)(x3 - x1)]/2+[(y3 + y2)(x2 - x3 )]/2-[(y1 + y2)(x2 - x1)]/2= [(y1 + y3)(x3 - x1)+(y3 + y2)(x2 - x3 )-(y1 + y2)(x2 - x1)]/2= (x3 y1 + x3 y3 - x1 y1 - x1 y3 + x2 y3 + x2 y2 - x3 y3 - x3 y2 - x2 y1 - x2 y2 + x1 y1 + x1 y2 )= [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]Thus the required area A of the triangular region is given byA = [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)] ------------------(1) Corollary

The points P1, P2 & P3 are collinear if A = 0*Trapezoid: A four sided shape in which no sides are parallel

II) Two and Three Straight Lines

I. Angle between two straight linesII. The equation of lines through the point of

intersection of two linesIII. Equation of the right bisectors & altitudes

of a triangleIV. Equation of straight lines in Matrix Form

A. One linear equation:B. A system of two linear equationC. A system of three linear equation


Let l1 : a1x +b1y + c1 = 0 & l2: a2x + b2y + c2 = 0 be two distinct lines.the slope of l1 is m1 = – a1/b1

And the slope of l2 is m2 = – a2 /b2

The lines are related to each other by one of the following forms:

l1 ║ l2

l1 ┴ l2

l1 & l2 are not related as l1 ║ l2 & l1 ┴ l2


I) Angle between two straight linesTheorem: let l1 & l2 be two non-vertical lines such that they are not perpendicular to each other. If m1 & m2 are the slopes of l1 & l2 respectively, then the angles from l1 to l2 is given by the formula.tan θ = (m1 – m2) / (1 + m2m1)

Corollary 1

l1 ║ l2 m1 = m2 Corollary 2

l1 ┴l2 m1m2 = -1 Corollary 3

the acute angle formed by two non-perpendicular intersecting lines l1 & l2 is given by the formulatan θ = (m1 – m2) / (1 + m2m1)


II) The equation of lines through the point of intersection of two lines:Let l1 : a1x +b1y + c1 = 0 ------------------------(1) & l2: a2x + b2y + c2 = 0 ------------------------(2)Be two intersecting lines.For a non-zero real number k, consider the equation

a1x +b1y + c1 + k (a2x + b2y + c2) = 0 ------------(3)Or (a1 + ka2 )x + (b1 + kb2)y + (c1 + kc2)We see that this equation is linear an hence represents a straight line, for different

values of k, (3) represents different lines. Thus (3) defines family of lines, one line of each value of k.

If (x1, y1) is the point of intersection of the lines (1) & (2) then, a1x1 +b1y1 + c1 = 0

& a2x1 + b2y1 + c2 =0From the above two equations, we havea1x1 +b1y1 + c1 + k(a2x1 + b2y1 + c2 ) = 0, since 0 + k . 0 = 0. this means that (x1, y1) lies on the

family of the lines (3). Hence, we conclude that (3) is the required family of lines through the point of intersection of (1) & (2)which is conformity with the result that an infinite number of lines can pass through a point.


III) Equation of the right bisectors & altitudes of a triangle:

Theorem 1:Right bisectors of a triangle are concurrent (Refers to figure 1(a))

Theorem 2:Altitudes of a triangle are concurrent (refers to figure 1(b))


VI) Equation of straight lines in Matrix Form:We know that a linear equation ax + by + c = 0 in two variables x & y where a, b & c are constants & a & b are not simultaneously zero represents a straight line. It is easy to solve two or three simultaneous linear equations by ordinary method. Howe ever, if the number of equation & variables become large, then it is very difficult & time consuming to solve them by an ordinary method. In such a case, it is convenient to write the given equation in matrix form & solved. We therefore describe briefly the method of transforming one, two or three linear equations to the matrix form. The method happens to be quite general and can be used in case of any number of equations.

VI) Equation of straight lines in Matrix Form

a) One linear equation:a linear equation

l: ax + by + c = 0 --------------------------(1)In two variables x & y can be written as

ax + by = –c In matrix form the above equation can be written as

[ax + by] = [c]

[a b] = [–c]

AX = KWhere A= [a b], X= & K = [–c]

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