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Submitted by: John Charlemagne Buan
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Submitted to: Ms. Harlene Santos
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Analytic geometry
From Wikipedia, the free encyclopedia

Analytic geometry, or analytical geometry, has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties. This article focuses on the classical and elementary meaning.
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and usesdeductive reasoning based on axioms and theorems to derive truth. Analytic geometry is widely used in physicsand engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, andsquares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. The numerical output, however, might also be a vector or a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.

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History
The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.[1] Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others.[2] Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes — by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead ofa priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.[3]
The eleventh century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra[4] with his geometric solution of the general cubic equations,[5] but the decisive step came later with Descartes.[4]
Analytic geometry has traditionally been attributed to René Descartes.[4][6][7] Descartes made significant progress with the methods in an essay entitled La Geometrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native French tongue, and its philosophical principles, provided a foundation for Infinitesimal calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descarte's masterpiece receive due recognition.[8]
Pierre Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse.[9] Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint. Fermat always started with an algebraic equation and then described the geometric curve which satisfied it, while Descartes starts with geometric curves and produces their equations as one of several properties of the curves.[8] As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree.

Basic principles

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Coordinates
In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered tripleof coordinates (x, y, z).
Other coordinate systems are possible. On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ. In three dimensions, common alternative coordinate systems includecylindrical coordinates and spherical coordinates.
Equations of curves
In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is said to be the equation for this line. In general, linear equations involving x and yspecify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.
Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x2 + y2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as theintersection of two surfaces (see below), or as a system of parametric equations. The equation x2 + y2 = r2 is the equation for any circle with a radius of r.

The distance formula on the plane follows from the Pythagorean theorem.
Distance and angle
In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the formula

which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula

where m is the slope of the line.
Transformations
| This section requires expansion.(December 2009) |
Transformations are applied to parent functions to turn it into a new function with similar characteristics. For example, the parent function y=1/x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote,and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if y = f(x), then it can be transformed into y = af(b(x − k)) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis. The b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like a, reflects the function in the y-axis when it is negative. The k and h values introduce translations, h, vertical, and k horizontal. Positive h and k values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.
Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.
Suppose that R(x,y) is a relation in the xy plane. For example x2 + y2 -1= 0 is the relation that describes the unit circle. The graph of R(x,y) is changed by standard transformations as follows:
Changing x to x-h moves the graph to the right h units.
Changing y to y-k moves the graph up k units.
Changing x to x/b stretches the graph horizontally by a factor of b. (think of the x as being dilated)
Changing y to y/a stretches the graph vertically.
Changing x to xcosA+ ysinA and changing y to -xsinA + ycosA rotates the graph by an angle A.
There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.
Intersections
While this discussion is limited to the xy-plane, it can easily be extended to higher dimensions. For two geometric objects P and Q represented by the relations P(x,y) and Q(x,y) the intersection is the collection of all points (x,y) which are in both relations. For example, P might be the circle with radius 1 and center (0,0): P = {(x,y) | x2+y2=1} and Q might be the circle with radius 1 and center (1,0): Q = {(x,y) | (x-1)2+y2=1}. The intersection of these two circles is the collection of points which make both equations true. Does the point (0,0) make both equations true? Using (0,0) for (x,y), the equation for Q becomes (0-1)2+02=1 or (-1)2=1 which is true, so (0,0) is in the relation Q. On the other hand, still using (0,0) for (x,y) the equation for P becomes (0)2+02=1 or 0=1 which is false. (0,0) is not in P so it is not in the intersection.
The intersection of P and Q can be found by solving the simultaneous equations: x2+y2 = 1
(x-1)2+y2 = 1
Traditional methods include substitution and elimination.
Substitution: Solve the first equation for y in terms of x and then substitute the expression for y into the second equation. x2+y2 = 1 y2=1-x2 We then substitute this value for y2 into the other equation:
(x-1)2+(1-x2)=1 and proceed to solve for x: x2 -2x +1 +1 -x2 =1
-2x = -1 x=½ We next place this value of x in either of the original equations and solve for y:
½2+y2 = 1 y2 = ¾

So that our intersection has two points:

Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, If we subtract the first equation from the second we get: (x-1)2-x2=0 The y2 in the first equation is subtracted from the y2 in the second equation leaving no y term. y has been eliminated. We then solve the remaining equation for x, in the same way as in the substitution method. x2 -2x +1 +1 -x2 =1 -2x = -1 x=½ We next place this value of x in either of the original equations and solve for y: ½2+y2 = 1 y2 = ¾

So that our intersection has two points:

For conic sections, as many as 4 points might be in the intersection.
Intercepts
One type of intersection which is widely studied is the intersection of a geometric object with the x and y coordinate axes.
The intersection of a geometric object and the y-axis is called the y-intercept of the object. The intersection of a geometric object and the x-axis is called the x-intercept of the object.
For the line y=mx+b, the parameter b specifies the point where the line crosses the y axis. Depending on the context, either b or the point (0,b) is called the y-intercept.
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Important themes of analytical geometry are * vector space * definition of the plane * distance problems * the dot product, to get the angle of two vectors * the cross product, to get a perpendicular vector of two known vectors (and also their spatial volume) * intersection problems * conic sections depending on the class, this may include rotation of coordinates and the general quadratic problems
Ax2 + Bxy + Cy2 +Dx + Ey + F = 0. If the Bxy term is considered, rotations are generally used.
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Vector space
From Wikipedia, the free encyclopedia
This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry).

Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2·w.
A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are best thought of as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.
Vector spaces are the subject of linear algebra and are well understood from this point of view, since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuityissues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.
Historically, the first ideas leading to vector spaces can be traced back as far as 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.
Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point (zero-dimensions), a line(one-dimension) and a solid (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.
When working exclusively in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a two-dimensional space, or in other words, in the plane.

Geometry
In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. The distance between (x1, y1) and (x2, y2) is given by:

Similarly, given points (x1, y1, z1) and (x2, y2, z2) in three-space, the distance between them is:

These formula are easily derived by constructing a right triangle with a leg on the hypotenuse of another (with the other leg orthogonal to the plane that contains the 1st triangle) and applying the Pythagorean theorem. In the study of complicated geometries,we call this (most common) type of distance Euclidean distance,as it is derived from the Pythagorean theorem,which does not hold in Non-Euclidean geometries.This distance formula can also be expanded into the arc-length formula.
Distance in Euclidean space
In the Euclidean space Rn, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norms, are sometimes used instead.
For a point (x1, x2, ...,xn) and a point (y1, y2, ...,yn), the Minkowski distance of order p (p-norm distance) is defined as: 1-norm distance | | 2-norm distance | | p-norm distance | | infinity norm distance | | | | p need not be an integer, but it cannot be less than 1, because otherwise the triangle inequality does not hold.
The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance.
The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance, because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets).
The infinity norm distance is also called Chebyshev distance. In 2D, it is the minimum number of moves kings require to travel between two squares on a chessboard.
The p-norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse.
In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body does not change with rotation.
Variational formulation of distance
The Euclidean distance between two points in space ( and ) may be written in a variational form where the distance is the minimum value of an integral:

Here is the trajectory (path) between the two points. The value of the integral (D) represents the length of this trajectory. The distance is the minimal value of this integral and is obtained when where is the optimal trajectory. In the familiar Euclidean case (the above integral) this optimal trajectory is simply a straight line. It is well known that the shortest path between two points is a straight line. Straight lines can formally be obtained by solving the Euler-Lagrange equations for the above functional. In non-Euclideanmanifolds (curved spaces) where the nature of the space is represented by a metric the integrand has be to modified to , where Einstein summation conventionhas been used.
Generalization to higher-dimensional objects
The Euclidean distance between two objects may also be generalized to the case where the objects are no longer points but are higher-dimensional manifolds, such as space curves, so in addition to talking about distance between two points one can discuss concepts of distance between two strings. Since the new objects that are dealt with are extended objects (not points anymore) additional concepts such as non-extensibility, curvature constraints, and non-local interactions that enforce non-crossing become central to the notion of distance. The distance between the two manifolds is the scalar quantity that results from minimizing the generalized distance functional, which represents a transformation between the two manifolds:

The above double integral is the generalized distance functional between two plymer conformation. is a spatial parameter and is pseudo-time. This means that is the polymer/string conformation at time and is parameterized along the string length by . Similarly is the trajectory of an infinitesimal segment of the string during transformation of the entire string from conformation to conformation . The term with cofactor is a Lagrange multiplier and its role is to ensure that the length of the polymer remains the same during the transformation. If two discrete polymers are inextensible, then the minimal-distance transformation between them no longer involves purely straight-line motion, even on a Euclidean metric. There is a potential application of such generalized distance to the problem of protein folding[1][2] This generalized distance is analogous to the Nambu-Goto action in string theory, however there is no exact correspondence because the Euclidean distance in 3-space is inequivalent to the space-time distance minimized for the classical relativistic string.

Algebraic definition
The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:[1]

where Σ denotes summation notation and n is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors [1, 3, −5] and [4, −2, −1] is:

Geometric definition[edit]
In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by . The dot product of two Euclidean vectors A and B is defined by[2]

where θ is the angle between A and B.
In particular, if A and B are orthogonal, then the angle between them is 90° and

At the other extreme, if they are codirectional, then the angle between them is 0° and

This implies that the dot product of a vector A by itself is

which gives

the formula for the Euclidean length of the vector.
Scalar projection and the equivalence of the definitions[edit]

Scalar projection
The scalar projection (or scalar component) of a Euclidean vector A in the direction of a Euclidean vector B is given by

where θ is the angle between A and B.
In terms of the geometric definition of the dot product, this can be rewritten

where is the unit vector in the direction of B.

Distributive law for the dot product
The dot product is thus characterized geometrically by[3]

The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar α,

It also satisfies a distributive law, meaning that

As a consequence, if are the standard basis vectors in , then writing

we have

which is precisely the algebraic definition of the dot product. More generally, the same identity holds with the ei replaced by any orthonormal basis.

The cross product of two vectors a and b is denoted by a × b. In physics, sometimes the notation a ∧ b is used,[2] though this is avoided in mathematics to avoid confusion with the exterior product.
The cross product a × b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
The cross product is defined by the formula[3][4]

where θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), ‖a‖ and ‖b‖ are the magnitudes of vectors a and b, and nis a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule as illustrated. If the vectors aand b are parallel (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.

The cross product (vertical) changes as the angle between the vectors changes
The direction of the vector n is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the picture on the right). Using this rule implies that the cross-product is anti-commutative, i.e., b × a = −(a × b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector.
Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction.
This, however, creates a problem because transforming from one arbitrary reference system to another (e.g., a mirror image transformation from a right-handed to a left-handed coordinate system), should not change the direction of n. The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector. See cross product and handedness for more detail.
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Names

According to Sarrus' rule, the determinant of a 3×3 matrix involves multiplications between matrix elements identified by crossed diagonals
In 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced both the dot product and the cross product using a period (a . b) and an "x" (a x b), respectively, to denote them.[5]
In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford coined the alternative names scalar product and vector product for the two operations.[5] These alternative names are still widely used in the literature.
Both the cross notation (a × b) and the name cross product were possibly inspired by the fact that each scalar component ofa × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a · b involves multiplications between corresponding components of a and b. As explained below, the cross product can be expressed in the form of a determinant of a special 3×3 matrix. According to Sarrus' rule, this involves multiplications between matrix elements identified by crossed diagonals.
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Computing the cross product
Coordinate notation

Standard basis vectors (i, j, k, also denoted e1, e2, e3) and vector components of a (ax, ay, az, also denoted a1, a2,a3)
The standard basis vectors i, j, and k satisfy the following equalities:

which imply, by the anticommutativity of the cross product, that

The definition of the cross product also implies that (the zero vector).
These equalities, together with the distributivity and linearity of the cross product, are sufficient to determine the cross product of any two vectors u and v. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors:

Their cross product u × v can be expanded using distributivity:

This can be interpreted as the decomposition of u × v into the sum of nine simpler cross products involving vectors aligned with i, j, or k. Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using the above mentioned equalities and collecting similar terms, we obtain:

meaning that the three scalar components of the resulting vector s = s1i + s2j + s3k = u × v are

Using column vectors, we can represent the same result as follows:

Matrix notation[edit]
The cross product can also be expressed as the formal[note 1] determinant:

This determinant can be computed using Sarrus' rule or cofactor expansion. Using Sarrus' rule, it expands to

Using cofactor expansion along the first row instead, it expands to[6]

which gives the components of the resulting vector directly.

The intersection of A and B is written "A ∩ B". Formally:

that is x ∈ A ∩ B if and only if * x ∈ A and * x ∈ B.
For example: * The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. * The number 9 is not in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of odd numbers {1, 3, 5, 7, 9, 11, …}.[2]
More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus,
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
Inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. Now the intersection of A and B may be written as the complement of the union of their complements, derived easily from De Morgan's laws:
A ∩ B = (Ac ∪ Bc)c