Vector Subtraction

Vector Subtraction is a special case of vctor addition. 
Let there are two Vectors A and B then subtraction of Vector B from A is A – B which can also be written as A + (-B).
So, A – B = A + (-B)
Here (–B) is a Vector that is parallel to Vector B but is in reverse or opposite direction of Vector B.
According to parallelogram law of addition, if there are two Vectors such that they form the two consecutive sides of a parallelogram then the sum of the Vectors is represented by the diagonal of the parallelogram passing through the point at which the tails of two Vectors meet. The geometrical representation of A-B is given in following diagram.

The above method is known as head to tail method.
You can find Subtraction of Vectors by using perpendicular components method. In this method you need to split each Vector into perpendicular components. We normally split the Vectors in x and y directions.
Suppose a Vector of magnitude 5 makes an angle of 30o with horizontal then its vertical component will be 5sin(30)j and horizontal component will be 5cos(30)i. i and j are unit Vectors along the x and y directions respectively. If a component is in downward or left direction assign a negative sign to it. Similarly Split all the given Vectors in such a way. Now add all the horizontal components together and add all the vertical components separately. Subtract a component if its magnitude is negative. Now calculate the resultant magnitude of the Vectors using Pythagoras theorem R2 = A2 + B2 . Here A is sum of horizontal components, B is sum of vertical components and R is resultant.
The angle that the resultant Vector makes with the horizontal is given as: Ө = tan-1|B|/|A|  . Here|A| and |B| are magnitudes of the Vectors.
Subtracting Vectors is similar to addition of Vector. The difference is just of the negative sign.
Let us see some Vector Subtraction Examples now:
Example 1) Subtract Vector B = 9i+7j+2k from A =3i + 5j + k.
Solution) To subtract B from A we write them as A-B = (3i + 5j + k) – (9i - 7j + 2k)
Now add the Vectors in the direction i, j, k separately. As the given Vectors are three dimensional we are considering z direction also. i, j, k refers to unit Vectors in x, y and z direction.
(3-9)i = -6i
(5- (-7))j= 12j
And (1-2)k = -k
A-B=(-6i+12j-k)
|A-B| = √(〖(-6)〗^2+〖12〗^2+〖(-1)〗^2 ) =√(36+144+1) = √181

Venn diagram

An illustrative representation of relationship between and among the given sets that have something in common are called Venn Diagrams. These diagrams are represented using circles, the most commonly used ones are the two circles and three circle Venn-diagrams. It is used mainly to depict the intersections of sets which mean to show the objects or things common to all the sets. The Picture of Venn diagram with two circles or three circles consists of a rectangle which has in it two circles or three circles respectively. Intersection of sets is shown as overlapping circles, if no common objects then the circles are shown separately.

The total number of elements in the set is shown at the right corner of the rectangle using a symbol ‘µ’ or ‘E’. The object or elements which do not belong to any of the sets is shown outside the circle within the rectangle. A glance at the Venn diagram help to understand the complete picture of the relationship among the sets given and also help to find the unknowns values using  the diagram.

Example of Venn Diagram is as follows: set X={1,2,3,5,7,9} and set Y={7,9,11,13}, the intersection of these sets given by X∩Y={7,9}.
The Venn-diagram showing the relationship between these two sets is as given below, the pink region shows the intersection of the sets, X∩Y.It also shows the elements only in the set X {1,3,5} in green color and the elements only in the set Y {11,13} in yellow color.

Venn diagram compare and contrast the relationship between the elements of the sets, they are basically used to visualize the relationship between two or three sets. They are also used to compare and contrast the characteristics of the elements of the sets.  An outline about the sets can be easily created using the Venn-diagrams. Venn Diagram with three circles is a type of diagram which involves three sets.  Let us learn more about this using a simple example: A survey was done on a group of children as how many like to play baseball, football and ice hockey shown by the Venn-diagram below,

The above Venn-diagram gives the following information about the three sets, F, B and I.

• Total number of children (E) would be sum of all the numbers on the three circles, 5+14+8+2+16+5+20=70, so E=70.
• n(F)=29, n(B)=47, n(I)=31
• only n(F)=5, only n(B)=20, only n(I)=16
• n(F∩B∩I)=8, n(F∩B)=22, n(B∩I)=13, n(F∩I)=10
• only n(F∩B)=14, only n(B∩I)=5, only n(F∩I)=2

Greatest common divisor

The numbers when multiplied together to get another number are called the factors of the number. For example consider the number fourteen, 2 when multiplied with 7 gives 14 and hence 2 and 7 are the factors of the number 14 which includes 1 and also 14. The factors of 6 are 1, 2, 3 and 6.
The factors of 9 are 1,3 and 9. The common factors of 6 and 9 are 1 and 3. So, we can define common factors as the factors which are common to all the numbers considered.

 Let us now learn about Greatest common factor which is also called Highest Common Factor and at times Greatest Common Divisor. It can be defined as the greatest integer which divides the number evenly that is without any remainder. In short it is called GCD.

For example, GCD of 12 and 18 is 6 as 6 is the greatest integer which divides 12 and 18 evenly. Let us now find the Greatest Common Divisor of the numbers 36 and 72. The first step to find the GCD would be to write all the factors of each of the numbers, factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36; factors of 72 are 1, 2, 3, 4, 6, 9, 12, 18, 36 and 72.

The next step is to identify the greatest integer which is common to both the lists, 36 is the largest integer common to both the lists and hence GCD of 36 and 72 is 36.

There are different methods to find the GCD of two or more numbers the simplest being the prime factorization method where prime factors of each numbers are listed and then the greatest prime factor common is determined.

The other method is Euclid’s algorithm which involves a process of repeated division given by the Greatest Common Divisor formula, gcd(a,b)=gcd(b, a mod b) where [a mod b] is got by { a – b[a/b]}, mod here means modulus. There are variations in the formula according to the values of a and b.

Let us now learn about properties of GCD which can be given as:

• For any integers ‘a’ and ‘b’ there exist integers ‘p’ and ‘q’ such that [p.a+q.b]=gcd(a,b)
• If ‘a’ and ‘b’ are relatively prime integers then there exist an integer ‘p’ such that pa=1(mod b)
• If ‘a’, ‘b’ and ‘c’ are relatively prime integers and if a divides bc then a divides c

Estimation and Rounding

Estimation and rounding is a important concept in basic mathematics. Number system is the concept that defines everything about numbers and estimation and rounding is a concept covered within number system.

Let’s have a look at both the concepts in this post.
The process of rounding a number estimating its nearest value is called estimation and rounding. For example: After she started breathing exercises during pregnancy , her medical issues lessen to about 59.8%. Here, we can also say the same thing in another way: After she started breathing exercises during pregnancy , her medical issues lessen to about 60%. This is rounding a number estimating to its nearest value. This is an approximate value of the given number.

There are certain rules to be followed while incorporating estimation and rounding, two are mentioned as below:
Rule 1: If the number that needs to be rounded is followed by a number which is greater or equal to 5, it can be rounded to its nearest greater number.

• Example 1: The hospital bag checklist of her NGO includes about 68.8% of things for baby and child care.
The hospital bag checklist of her NGO includes about 70% of things for baby and child care.

• Example 2: These are my most comfortable shoes ; my feet stays almost 95% relaxed wearing the same.
These are my most comfortable shoes; my feet stays almost 96% relaxed wearing the same.
Rule 2: If the number that needs to be rounded is followed by a number which is less than 5, it can be rounded to its nearest smaller number.

• Example 1: She bought apples for 61 rupees per kg for her daughter.
She bought apples for 60 rupees per kg for her daughter.

• Example 2: I have gained almost 6.3 kg of weight after getting married.
I have gained almost 6 kg of weight after getting married.
These are the basics about estimation and rounding.

Finding Determinant Value of 3x3 Matrices

Matrices, like any other expressions,are also subjected to basic algebraic operations. Multiplication is one of those. In addition to normal restrictions on matrix operations, multiplication of two matrices has additional restrictions and the method is a bit strange.

Suppose A and B are two matrices, their product is defined only if the number of columns of A and the number of rows of B must be same. Else, the multiplication is not defined. The algorithm for multiplying 3x3 matrices is described below.

The first element of first row and first column of the product C of matrix A and matrix B is the sum of the products of the elements of the first row with the corresponding elements of the first column of B.

For the second element in the same row, the products are now first row of Avs second column of B. Like this the entire elements of C are found.

With the restrictions and strange algorithm of the process, it can clearly be concluded that multiplication of matrices is not commutative.

A matrix just conveys the details of the data and it does not give any solution to any question on the relations between the items. On the other hand if the same arrangement of data indicates an operation for a solution, then that particular arrangement is called the determinant of the same matrix.

A matrix is denoted by containing the items of a data within a set of elongated square brackets. In a determinant the same are enclosed by two thin vertical bars (similar to absolute value symbol). The following is an example.

The determinant of the matrix indicates the solution by the operation (a1 *b2- a2*b1).
The determinant of a matrix with 3 rows and 3 columns is called a determinant 3x3 matrix or just as determinant 3x3.
To find determinant of 3x3 matrix, first rewrite the elements in the same order but with the symbol of a determinant. To find the value of determinant of a 3x3, the following is the procedure.
Consider a11, the first element in the first row and column. By deleting the row and column that includes a11, you are left with a 2 x 2 determinant which is called as the minor of the element a11and denoted as, M11.
Same way, for the second and third elements in the first row a12 and a13 , the minors are M12 and M13 respectively. Now the formula for the value of the determinant D is,
D = (a11*M11  -a12*M12 + a13*M13)

Set theory

We deal with many types objects in mathematics and when we combine them and out them together, we call them Sets Math and study of that collection of objects is the theory of sets.

The objects of the sets are usually called elements.

Set Theory Notation– We know that the collection of elements is termed as sets. We usually write a st in curly brackets.
For example: - A st of first five even numbers can be written as {2, 4, 6, 8, and 10} or the saet of first seven natural numbers is written as {1, 2, 3, 4, 5, 6, 7}.

The intersection of two sets is symbolized by inverted U and the union of two sets is written as with symbol U. Union of sets A and B means the saet of all objects that is member of both the sets and this can be written as A U B.

The intersection of two sets that is A intersection B is all the elements that are part of both A and B. If A is contained or part of B than we can call A as subset of B or if B has A contained in it then we can call B as superset of A.

We use different types of axioms in the theory of sets. Axioms can be logical or non-logical. Logical axioms tells us the universal truth which is like a + b = b + a.

An axiom act as a starter to derive the proofs similarly in Set Theory Proofs, we make use of Axiomatic Set Theoryto derive various proofs. 

There are many types of axioms that are used in the axiomatic theory which are axiom of existence, axiom of equality, axiom of pair, axiom of separation, axiom of union, axiom of power, axiom of infinity, axiom of image, axiom of foundation and axiom of choice.

Set Theory Problems are simplified using the axioms of the theory.

We have different types of special sets that we use in mathematics and that are used in the theory of sets.
For example the saet of all integers is denoted by Z where Z = {…-3, -2, -1, 0, 1, 2, 3…} and st of all prime numbers is denoted by P where P = {2, 3, 5, 7, 11, 13, 17, 19, 23…}

LCM (least common multiple)

LCM is an abbreviation for Least Common Multiple’. It is also called as ‘lowest common multiple’ and less popularly as ‘smallest common multiple’. For a given set of two or more integers, the least common multiple is the least integer that can be divided by all the integers in the given set.

The fundamental method of finding the smallest common multiple for a set of integers is first list out the multiples of each integer. Pick up the common integers that appear in all such lists. Now figure out the lowest among them. It is the least common multiple. For example, let us try to find the least common multiple of 4 and 6.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 …
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42 …
The integers that are seen common in both the lists are: 12, 24, 36 …
The smallest integer among these commons is 12.
Therefore, the smallest common multiple of 4 and 6 is 12.

There is another method to find the lowest common multiple by using the following formula.
LCM of x, y = (x*y)/(GCF of x and y)
Let us try this method to find the least common multiple of 12 and 16.
The GCF of 12 and 16 is 4.
Therefore, the least common multiple of 12 and 16 = (12*16)/(4) = 3*16 = 48.

In cases where you need to find the lowest common multiples of many numbers, especially large, you can find that by prime factorization. The method here is to do the prime factorization of each number. The lowest common multiple is the product of the highest power of each prime number, out of all factors. Let us find the least common multiple of 9 and 12 by this method.
The prime factorization of 9 is 3*3 = 32.
The prime factorization of 12 is 3*4 =3*2*2 = 3* 22.
The product of the highest powers of prime numbers is 32* 22= 9*4 = 36.
Hence, the smallest common multiple of 9 and 12 is 36.

The knowledge of lowest common multiple for a given set of integers are greatly helpful in finding lowest common denominator for a given set of fractions. The lowest common denominator is the lowest common multiple of the denominators of the given fractions. Therefore, LCM problems are mostly found in such applications. For example the lowest common divisor of the fractions (1/3) and (1/5) is 15, which is the lowest common multiple of 3 and 5.