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

Definition of Plane

Plane Shape

Examples

History

Planes in fiction

Applications

Two-dimensional coordinate system Three-dimensional coordinate system

Orientation and handedness

Area of Plane Shapes

Equations of plane

Definition of Plane

A plane is a flat surface with no thickness.

It extends forever .

A plane is a flat surface with 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 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

Equations of straight lines

Two and Three Straight Lines

I) Equations of straight lines

The slope of a Line

The slope of a Line Joining Two Points

Equations of a straight line Parallel and perpendicular to coordinate axes

Derivation of Standard Forms of Equations of Straight Lines

A linear Equation in Two variables represents a Straight Line

To Transform the General Linear Equation in Standard Forms

Position Of a Point with respect to a Line:

Distance of a point from a Line:

Distance Between Two Parallel Lines:

Area of a Triangular Region Whose Vertices are Given:

The slope of a Line

Let l be any non vertical line in the plane. Let P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) be any distinct points on l as shown in figure. If we move along l from P 1 to P 2 we move y 1 - y 2 units in the y - direction and x 1 -x 2 units in the x- direction.

Rise Run P1(x1, y1) P1(x2, y2) y1-y2 x1-x2 x y l o

The slope of a Line Joining Two Points

The 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 P 1 M & P 2 N on x-axis & perpendicular P 1 P 3 on P 2 N. Then angle P 2 P 1 P 3 = Ф , P 1 P 3 =x 2 -x 1 . In triangle P 1 P 3 P 2 , we have m=y 2 -y 1 /x 2 -x 1 =tan Ф ( Refers to figure Case II)

c) Case III When 90°< Ф <180°

tan ( π - Ф )= y 2 -y 1 /x 1 -x 2

- tan Ф = y 2 -y 1 /x 1 -x 2

( Refers to figure Case III)

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

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)

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)

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)

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)

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

There are six standard forms of equations of

a) Slope Intercept Form

b) Point-Slope Form

c) Two-Points Form

d) Two Intercepts Form

e) Normal Form

f) Symmetric Form

a) Slope Intercept Form

if 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 Form

Theorem: An equation of non –vertical line l with slope m that passes through the point

P(x 1 , y 1 ) is given by

y-y 1 =m(x-x 1 )

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)

c) Two-Points Form

Theorem: An equation of non-vertical line l passing through two points P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) is given by

y-y 1 =(y 2 -y 1 )(x-x 1 )/(x 2 -x 1 )

This is called Two-Points Form of a line

Two Intercepts Form

Theorem: An equation of a line l whose non-zero x & y intercepts are a & b respectively, is given by

x / a +y / b=1

This is called Two-intercepts Form of a line

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

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

Symmetric Form

Theorem: An equation of non-vertical line l with inclination Ф passing through a point P(x 1 , y 1 ) is given by

x-x 1 /cos Ф = y-y 1 /sin Ф =r

Directed distance from P(x 1 , y 1 ) to any point P’(x, y) on l

This is called symmetric form of the line

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

A linear Equation in Two variables represents a Straight Line

Every linear equation of two variables is given by

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

Case I

If a ≠ 0, b = 0

Then equation 1 becomes

ax + c = 0

x = -c/a

Which is the equation of a straight line parallel to y-axis at a directed distance –c/a from y-axis

Case II

If a = 0, b ≠ 0

by + c = 0

y=-c/b

Which is the equation of a straight line parallel to x-axis at a directed distance –c/b from x-axis

Case III

If a ≠ 0, b ≠ 0

ax + by + c = 0

by = -ax -c

y = - ax/b - c/b

Which is the slope-intercept form of a straight line with slope m = -a/b & k = -c/b

To Transform the General Linear Equation in Standard Forms:

Slope Intercept Form

Point-Slope Form

Two-Points Form

Two-Intercepts Form

Normal Form

Symmetric Form

vi) To Transform the General Linear Equation in Standard Forms

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

Slope Intercept Form:

Equation 1 can be written as

y= – ax/b – c/b = mx + k

where slope m = –a/b & k= – c/b which is the required transformed slope intercept form of equation 1

Point-Slope Form:

The slope of a line ax + by + c = 0 is m = –a/b

A point on the line is (- c/a, 0)

Therefore the transformed equation is

y – 0 = – ax/b – [ x – ( – c/a)]

y = – ax/b – ( x + c/a)

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)

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

Normal Form:

We know that an equation of a line in a normal form is

x cos Ф + y sin Ф = p

Let r cos Ф = a & r sin Ф =b

Then

r 2 cos 2 Ф =a 2 & r 2 sin 2 Ф =b 2

so r 2 cos 2 Ф + r 2 sin 2 Ф = a 2 + b 2

r 2 ( cos 2 Ф + sin 2 Ф ) = a 2 + b 2

r 2 = a 2 + b 2

r = ± √ a 2 + b 2

Symmetric Form:

We have sin Ф = b/ ± √ a 2 + b 2 , cos Ф = a/ ± √ a 2 + b 2

a point on ax + by + c = 0 is ( - c/a, 0)

Thus the equation in the symmetric form becomes

[{ x – ( - c/a )} / a] / ± √ a 2 + b 2 = (y – 0 /b) / ± √ a 2 + b 2 = r

Which is the required transformed equation. Sign of radical is to be properly choosen.

Position Of a Point with respect to a Line:

(Refers to figure 1 (a)

Theorem: Let P(x 1 , y 1 ) 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

Above the line l if

ax 1 + by 1 + c > 0

Below the line l if

ax 2 + by 2 + c < 0

(Refers to figure 1 (b)

Corollary I

The point P(x1, y1) is above or below the line l: ax + by + c = 0 if

ax 1 + by 1 + c & b have the same sign or have opposite sign respectively

Corollary II

The point P(x1, y1) and the origin O(0, 0) are:

On the same side of l if ax 1 + by 1 + c & c have the same sign

On the opposite side of l if ax 1 + by 1 + c & c have the opposite sign

l O x y Figure 1(a) l O x y Figure 1(b) P(x1, y1) P1(x1, y') P(x1, y1)

Distance of a point from a Line:

Theorem: The distance d from the point P(x 1 , y 1 ) to the line

l: ax + by +c = 0 is given by

d = │ ax 1 + by 1 + c │ / √ a 2 + b 2

Corollary I

If l is horizontal, then

d = │ by 1 + c /a │

Corollary II

If l is vertical, then

d = │ ax 1 + c /a │

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

Distance Between Two Parallel Lines:

Let l 1 & l 2 be two parallel lines & P(x1, y1) be & y point on l 1 . Draw a perpendicular from P on l 2 which meets it at the point Q(x2, y2)

The distance from l 1 to l 2

(or from l 2 to l 1 ) 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.

l O x y Figure 1 P(x1, y1) Q(x2, y2) d 1 l 2

Area of a Triangular Region Whose Vertices are Given:

Let ABS be the triangle whose vertices are P 1 (x 1 , y 1 ),

P 2 (x 2 , y 2 ) & P 3 (x 3 , y 3 ) . Draw perpendicular P 1 A, P 2 C & P 3 B on x-axis

Area of the triangular region P 1 P 2 P 3

= Area of the trapezoidal* region P 1 ABP 3 + Area of the trapezoidal region P 3 BCP 2 - Area of the trapezoidal region P 1 ACP 2

= [( │ P 1 A │ + │ P 3 B │ )( │ AB │ )]/2 + [( │ P 3 B │+│ P 2 C │ )( │ BC │ )] –

[( │ P 1 A │+│ P 2 C │ )( │A C │ )]/2

= [(y 1 + y 3 )(x 3 - x 1 )]/2+[(y 3 + y 2 )(x 2 - x 3 )]/2-[(y 1 + y 2 )(x 2 - x 1 )]/2

= [(y 1 + y 3 )(x 3 - x 1 )+(y 3 + y 2 )(x 2 - x 3 )-(y 1 + y 2 )(x 2 - x 1 )]/2

= (x 3 y 1 + x 3 y 3 - x 1 y 1 - x 1 y 3 + x 2 y 3 + x 2 y 2 - x 3 y 3 - x 3 y 2 - x 2 y 1 - x 2 y 2 + x 1 y 1 + x 1 y 2 )

= [x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 3 (y 1 – y 2 )]

Thus the required area A of the triangular region is given by

A = [x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 3 (y 1 – y 2 )] ------------------(1)

Corollary

The points P 1 , P 2 & P 3 are collinear if A = 0

*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

Angle between two straight lines

The equation of lines through the point of intersection of two lines

Equation of the right bisectors & altitudes of a triangle

Equation of straight lines in Matrix Form

One linear equation:

A system of two linear equation

A system of three linear equation

Let l 1 : a 1 x +b 1 y + c 1 = 0 & l 2 : a 2 x + b 2 y + c 2 = 0 be two distinct lines.

the slope of l 1 is m 1 = – a 1 / b 1

And the slope of l 2 is m 2 = – a 2 / b 2

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

l 1 ║ l 2

l 1 ┴ l 2

l 1 & l 2 are not related as l 1 ║ l 2 & l 1 ┴ l 2

Angle between two straight lines

Theorem: let l 1 & l 2 be two non-vertical lines such that they are not perpendicular to each other. If m 1 & m 2 are the slopes of l 1 & l 2 respectively, then the angles from l 1 to l 2 is given by the formula.

tan θ = (m 1 – m 2 ) / (1 + m 2 m 1 )

Corollary 1

l 1 ║ l2  m 1 = m2

Corollary 2

l 1 ┴ l2  m 1 m2 = -1

Corollary 3

the acute angle formed by two non-perpendicular intersecting lines l1 & l2 is given by the formula

tan θ = (m 1 – m 2 ) / (1 + m 2 m 1 )

l x y Figure 1 1 l 2 Φ Φ 2 1 θ O

The equation of lines through the point of intersection of two lines:

Let l 1 : a 1 x +b 1 y + c 1 = 0 ------------------------(1)

& l 2 : a 2 x + b 2 y + c 2 = 0 ------------------------(2)

Be two intersecting lines.

For a non-zero real number k, consider the equation

a 1 x +b 1 y + c 1 + k (a 2 x + b 2 y + c 2 ) = 0 ------------(3)

Or (a 1 + ka 2 )x + (b 1 + kb 2 )y + (c 1 + kc 2 )

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,

a 1 x 1 +b 1 y 1 + c 1 = 0

& a 2 x 1 + b 2 y 1 + c 2 =0

From the above two equations, we have

a 1 x 1 +b 1 y 1 + c 1 + k(a 2 x 1 + b 2 y 1 + c 2 ) = 0, since 0 + k . 0 = 0. this means that (x 1 , y 1 ) 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.

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))

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)

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