Coordinate geometry

Co-ordinate geometryCoordinates are a set of values that show an exact position. Co ordinate geometry is that branch of geometry in which two real numbers are used to indicate the position of a point in a plane.

Directed linesA directed line is a straight line with number units positive, zero and negative.

There are 3 main expressions in determining coordinate geometry-

The coordinates of the origin are (0,0).

The coordinates of any point on X-axis is (X,0)say (5,0) if the point is +5 units on X-axis from the origin on right hand side towards the direction of the arrow.

The coordinates of any point on y-axis is (0,y) say (0,-5) if the point is -5 units on the y-axis from the origin downwards on the vertical axis.

Quadrants-The two directed lines when they intersect at right angles at the point of origin, divide their plane into four parts or regions.

Co-ordinates of midpoint-We can find out the coordinates of a mid point from the coordinates of the any two points using the following formula:

Xm=X 1+X 22

; Ym=Y 1+Y 22

Distance between two points-

The distance between two points P (X1,Y1) and Q (X2,Y2) can be defined by

d=√¿¿

Section Formula- In case of internal division- X= ((mx2 + nx1)/(m + n), Y=(my 2 + ny1)/(m + n)).In case of external division- X= ((mx2 - nx1)/(m - n), Y=(my2 - ny1)/(m - n)).

Coordinates of a centroid-

X=x1+x2+x 3

3

Y=y1+ y 2+ y 3

3

Straight line-Mathematically, it is defined as the shortest distance between two distinct points.

Quadrants- Any of the 4 equal areas made by dividing a plane by an x and y axis.

Cartesian coordinate system -A coordinate system in which the coordinates of a point are its distances from a set of perpendicular lines that intersect at an origin, such as two lines in a plane or three in space.

Polar coordinate system-In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.

Co ordinate geometry is a very powerful tool and analytical process in decision making. The importance of coordinate geometry are given below-

1. The coordinate geometry is an important branch of mathematics. It mainly helps us to locate the points in a plane. Its uses are spread in all fields like trigonometry, calculus, dimensional geometry etc.

2. The construction field mainly uses the coordinate geometry.

3. The sketch of the building is pure geometry.

4. For printing pdf files we use this geometry.

5. For finding the distance between the places we use coordinate geometry.

6. For estimating profit and loss of a business geometrical equations are largely used.

7. For the determination of linear relationship between unit and production co-ordinate geometry is highly used in business.

Determinants

Determinant - The determinant of a matrix is a special number that can be calculated from the matrix. It tells us things about the matrixes that are useful in systems of linear equations, in calculus and more.

Properties of the determinant –

The determinant has many properties. Some basic properties of determinants are:

1.|At|= |A|

The determinant of matrix A and its transpose At are equal.

2. |A|= 0

3. A triangular determinant is the product of the diagonal elements.

4. If a determinant switches two parallel lines its determinant changes sign.

5. If the elements of a line are added to the elements of another parallel line previously multiplied by a real number, the value of the determinant is unchanged.

6.If a determinant is multiplied by a real number, any line can be multiplied by the above mentioned number, but only one.

7. If all the elements of a line or column are formed by two addends, the above mentioned determinant decomposes in the sum of two determinants.

8. |A·B| =|A|·|B| The determinant of a product equals the product of the determinants.

Cramer's rule-

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

Cramer's Rule says:

x = Dx/D

y = Dy/D

z = Dz/D

Working these out, gives:

D = 354Dx = 2652Dy = 432Dz = 480

x = 2652/354 = 7.4915257y = 432/354 = 1.2203389z = 480/354 = 1.3559322

Sarrus' rule or Sarrus' scheme-

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus.

Matrix

01. Introduction of Matrix –

A matrix is a rectangular array of number, symbols or expressions arranged in rows and columns.

A matrix is usually denoted by a capital letter and its elements by corresponding small letters followed by two suffixes, the first one indicating the row and the one indicate column.

Types Of Matrices-

Matrices can be of many types such as:

Square Matrix

Diagonal Matrix

Null Matrix

Scalar Matrix

Unit Matrix

Symmetric Matrix

Equal Matrix

Skew Matrix

Row Matrix

Colum Matrix

Sub Matrix

Definition Of Various Matrices –

Equal Matrix :Two matrices are said to be equal if and only if

a. They are of the same order

b. Each element of one is equal to the corresponding element of the other

Square Matrix: A matrix is said to be square matrix if its number of rows is equal to the number of columns, that is if m=n it is called square matrix.

Diagonal Matrix: A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

Null Matrix: A matrix in which every element is zero is called null matrix .

Scalar Matrix: A diagonal matrix whose diagonal elements are all equal is called a scalar matrix.

Unit Matrix: A square matrix where diagonal elements are one and remaining elements are zero is called a unit matrix. Such matrix is denoted by In.

Symmetric Matrix : A matrix is said to be symmetric, if it is A square matrix

A aij = aij {i , j } element is the same as the {j, i} th element.

Skew Matrix : A square matrix is said to be a skew matrix if

aij = -aij .

Row Matrix: Row matrix is a matrix with only one row .

Column Matrix: A matrix which has only one column is called column matrix.

Sub Matrix: A sub matrix of a matrix is obtained by deleting any collection of rows or columns.

Determinants:

It may be noticed that in each case a 2 by 2 determinant has been taken by omitting row and column of a particular row element in order a1,b1 and c1. Another thing to note is the alternating signs for this row element.

Minors of a Matrix :

Co-factors of a Matrix:

Adjoint Matrix:

In linear algebra, the adjoint (occasionally referred to as adjunct) of a square matrix is the transpose of the cofactor matrix.

Inverse of a matrix :

Importance of Matrix in Business-

1. Various situations in business and economics can be represented using matrices. This can be done using the following examples.

Annual productions of two branches selling three types of items may be represented.

2. Number of staff in the office can be represented.3. The unit cost of transportation of an item from each of the three factories to each of the four warehouses can be represented.

MappingIn modern mathematics, the equivalent expression for a function is mapping.

DEFINITION: If f is the rule which associates every element of set X with one and only one element of set Y, then the rule f is said to be the function or mapping from the set X to the set Y. This we write symbolically as f :X Y

If y is the element of Y, corresponding to an element x of X, given by the rule f, we write this as yfx or y=f(x) and read as “ y is the value of f at x”

Limit of a function

The limit of a function is that fixed value to which a function approaches as the variable approaches a given value. The function approaches this fixed constant in such a way that the absolute value of the difference between the function and the constant may be made smaller and smaller than any positive number, however small. This difference continues to remain less than this assigned number say –

“When the variable approaches still nearer to the particular value chosen for it.”