Learning About the Field of Vectors

The quantities in mathematics can have different features. The features can be magnitude and sometimes direction as well. All the quantities do not have the same feature. This difference in features can be the difference between the quantities. The quantities that have only magnitude are known to be scalar quantities and the quantities that have both the magnitude and a direction can be called vectors or carriers.

This is the basic difference between the two types of quantities. This difference must be understood to understand the difference between the two types of quantities. This difference can be very helpful in understanding the difference between the quantities. The vector space over a field can be understood only when the concept of carriers is clear and understood. It is part of a mathematical in which the carriers which can be multiplied with a scalar.

The operation of addition is also possible within this space. The scalars can be any numbers. Sometimes they can also be real numbers or can also be complex numbers. The complex numbers have an imaginary part attached to them. The flux of vector field is nothing but a flow property of carriers. This property is concerned with a unit area.

The flow is per unit area. This concept is very important in the field of physics also. Both mathematics and physics are related to each other. So, this concept appears in both physics and mathematics. The knowledge of physics will help in understanding of the concept. The mathematical plot vector field will explain the concept.

The pictorial representation of the concepts will help in the understanding of the concepts. So, the pictorial representations are always helpful.

The flow of a vector field is related to the flux of the field. These both concepts are interrelated and the understanding of one concept will help in the understanding of the other concept. The plot can be shown in the Cartesian plane using the ‘x’ and ‘y’ axes.

The vector field graph shows the orientation of the various carriers in the Cartesian plane. The carriers are shown aligned in the different directions. So, the orientation of the carriers can be easily understood with the help of this graph.

This graph shows the different directions associated with the carriers. The carriers are shown oriented in different directions and this can be understood from the graph. So, the graph can be very useful.

Introduction to Linear Algebra

Linear Algebra is one of the branches of Mathematics.  These helps with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are found in modern math thus, it is widely used in both abstract algebra and functional analysis.
It is also found in solving problems in analytic geometry.
 Its application is found extensively  the natural sciences and the social sciences.  This forms an intro to Linear Algebra.

 The mathematical approach of solving for variables in equations is Algebra.  
It does the manipulation of these variables using mathematical structures.
It is like taking algebra to another level by manipulating these variables using mathematical structures. This also helps to write the entire system of equations in the form of matrices. When expressed in the form of matrices this to study in depth of the individual parts of matrix which is known as vertex. A vertex is simply a representation of a coordinate.

Linear algebra is a branch of mathematics.  The creation of analytic geometry led to  the system of linear equations  acquire a new significance.
Linear-algebra is important for both pure mathematics and applied mathematics.    
Linear form of algebra when combined with calculus facilitates the solution of linear systems of differential equations.

Analytic geometry, engineering, physics, natural sciences, computer science, and the social sciences   use the techniques of Linear-algebra.
Linear mathematical models are sometimes used to approximate Non-linear mathematical models because linear algebra is such a well-developed theory.
The real world applications need Linear-Algebra.  Algebra is all about linear system of equations and their solutions Linear-Algebra usually consists of the linear set of equations as well as their transformations on it.  Linear Equations  consists of  topics  like  Linear Equations , Matrices, Determinants, Complex numbers, Second degree equations, Eigen values ,Linear Transformations.

Linear algebra help to mathematics: It is a useful branch of mathematics.   Below listed is some of the applications of Linear form of Algebra.  Constructing curves, Least square approximation, traffic flow, Electrical circuits, Determinants, Graph theory, Cryptography.
It   is used to draw graphs. Most equations of linear algebra will   be a straight line. For example draw a graph for a vehicle travelling at a constant speed at various time intervals.
The graph helps in determining the    unknown variable that is the distance by plotting in on the graph.  This can also be used for   a multitude of different functions that is a ready tool for  lots of different real life functions.

Learning to Solve Algebraic Equations

There are various disciplines in the field of mathematics. It is a very interesting subject to learn and can be fun. To simplify algebraic equations, the concept of equations has to be clear. In equations there are both constants and variables present.

There is difference between both these. The value of constants almost remains same throughout but the value of variable changes. Equations can be solved with the help of various methods. The method of elimination can also be used.

It is very simple to use and can be easily learnt. One of the variables is eliminated to find the value of the other variables in this method. The equation algebra can be an interesting subject to learn if one has interest to solve equations. Sometimes equations in algebra can be very tricky to solve and one has to be careful while solving them.

The concept equations algebra is nothing but the equations that are present in this discipline of mathematics. There can be various methods to solve these equations. Many methods are available online to solve them. These methods can be easily learnt and applied while solving them. The algebra equations online will help in explaining the concept better.

Once the equations are solved, various values of the variables present in the equations are obtained. These values must satisfy the whole equation; otherwise they cannot be taken as the solution of the equation. These solutions must be feasible and must be acceptable.

An equation can have more than one solution. Once the solutions are obtained, it can be checked whether the solutions are right or not. Sometimes even complex numbers can be obtained as part of the solutions. In a complex number there is an imaginary part attached to it. There is also a real part also to the complex number. Both the real and imaginary part forms the complex number.

The art of solving equations can be very interesting. The equations can be of different types. There can be linear and quadratic equations. The difference between the two is only in their degree. The linear equation is very simple to solve compared to the quadratic equation.

The linear equation can be easily solved with the help of basic arithmetic operations and the principle of transposing. Once these concepts are known the linear equations can be easily solved. Once solved the solutions could be easily checked.

Boolean Algebra Tutorial

This Boolean Algebra Tutorial gives us a holistic view of what Boolean functions and calculations are and will surely aid in solving many mathematical problems related to Boolean algebra
Basic operations in Boolean algebra are as follows:
Conjunction: also denoted by an inverted V this operation basically is similar to the intersection operation. In this we have a specific formula. Actually this formula is derived by basic algebra intersection operation:
A intersection B = A + B – A V B - - - - - - -(1)
Here V is the union operation.
In Boolean Algebra we only deal with true or false here. So anything that is not true will be a false. We can also denote it by zero and one. Here one will denote a true and zero will denote a false.
Now let us state the complimentary operation. This is also called as the negation operation. It is a singular operand operation. That is in this we will need just one operand to calculate the result. Let us take an example. Say A = True. Now negation A will be false. This operation can be stated as :
Let A = True;
~ A
Output A. Now the value of A will be false.
Union operation has already been discussed. In Boolean we have certain Boolean Algebra Rules to be followed for the same. Let us take a quick look at them.
True union True = True
(True union False) or (false union true)= True
False union False = false

Similarly for intersection we have
True intersection true = true
True intersection false = false
False intersection false = false
As we can see both the operation are just opposite to each other. These are Boolean form of algebra rules. Let us state them in a more simplified way.
A + 1 = 1
A + 0 = A
A . 1 = A
A . 0 = 0
So it is all about the multiplication or the dot product and the addition with zeros and ones. A here can have any value that is one or zero. Boolean Algebra Examples can be:
Let a = 1
B = 0 so a . b = 1 . 0 = 0
Similarly let b = 0, ~ b = 1. This is an example of complimentary operation.
Other rules for the Boolean forms are:
~(~A ) = A
A + ~A = 1
A. ~A=0
~(A+B) = (~A) .(~B)
~(A.B) = (~A) + (~B)

Help your Child to Score Better in Exam with Online Tutoring

Online tutoring helps students to attain pinnacle of success in the academic field. Get a one-on-one learning session from a preferred tutor any time just by using a computer and internet connection. This learning mode is a great way to overcome student's exam anxiety.


Online tutoring is the ideal learning resource as it helps students to score good marks in exam. It has been observed that parents are more concerned about their child's education and usually they impose lot of pressure on them to do well in exam. The high expectations of parent sometimes create an exam anxiety in student, which leads to poor performance. To attain academic goal, a student should opt for extra learning resource. Online learning is a wonderful option to get a thorough knowledge on a specific topic. In a virtual classroom, students get personalized attention from a tutor, which is extremely comfortable and secured way of learning.

It is apparent that while studying in classroom, many students come across doubts but they don't feel like resolving it from the tutor due to hesitation and embarrassment. In a secure web environment, students have the privilege to clear any type of doubt any time from their online tutors. Besides this, students also get assignment and homework assistance in a step-by-step manner. Experience a streamlined and interactive mode of learning that will surely improve your learning skills, makes you understand a particular topic and also guide you during examination time.

Exam is the yardstick that measures the academic performance of every student therefore it is important to get a good learning guidance. Online tutoring not only caters to the educational requirement of students but also make them efficient while dealing with difficult questions. These one-one-one learning sessions are channelized through a personal computer. A student can get an instant connection with a tutor by log-in to a tutoring website. Apart from this, every topic is well explained by the tutor with the help of a whiteboard. Moreover, to make the session more active, a student can also ask question to the tutor through chat.

Classroom sessions are undoubtedly important but acquiring knowledge from additional learning resource is extremely beneficial for students of all grades. Attending classes on regular basis help students to understand a concept in a better manner. But learning a subject in a web environment will absolutely make you confident during exams. No matter how much time students spent in their studies, learning a subject with strong concentration can actually help you out to understand and remember each and every concept. With online learning, a student can obtain comprehensive knowledge on a particular topic and can perform better to attain academic objective.

Vector Components

As we know Vector is a quantity which has some magnitude and direction. Any Vector can be split into two Vector components such that the two Vector  component are perpendicular to each other.
For example as shown in the figure given below a Vector in the north west direction can be split into two components which are in north and west direction. Similarly a Vector in north east direction can be divided into two components which are in north and east direction. Note that the magnitude of these two components will not be same as that of the initial Vector.

Vector Components and the Vector which is divided into components together form a right angled triangle such that the two components are base and altitude of the triangle and original Vector if hypotenuse of the triangle. Given below a Vector ‘a’ which is divided into two components ax and ay such that ax, ay, a form the base height and hypotenuse of the triangle. ay component shown in figure can be shifted in right direction parallel to form the height of the triangle.  
Components of a Vector can be found out by using properties of trigonometry which are based on a right angled triangle.

According to trigonometry, in a right triangle if hypotenuse makes an angle Ө with the base then,
sinӨ = height or perpendicular/hypotenuse
cosӨ = base/hypotenuse.
For previous illustration sinӨ = ay/a
cosӨ = ax/a
So, Vector Component ax and ay will be:
ax = a(cosӨ)   ……….(component of Vector ‘a’ in x direction)
ay = a(sinӨ)  ………….(component of Vector ‘a’ in y direction)
Vector a can be written as: a = axi + ayj = a (cosӨ)i + a (sinӨ) j
If a Vector is given as a = a1i + a1j, then the Vector components can be read directly. Here a1 is component in x (horizontal) direction and a2 is Vector in y (vector) direction.
Let us take an example of Components of Vectors :
Example) Find the component of a Vector with magnitude 5 and makes an angle of positive 60o with horizontal.
Solution) let the Vector be P.
components of Vector P in x direction will be: Px= PcosӨ
= 5cos(60o)
= 5(1/2)
= 5/2 = 2.5
components of Vector P in y direction: Py= P(sin60o)= 5(√3/2)
So Vector P = 2.5i +5(√3/2)j
Note) If the Vector makes 60o with the vertical then Py = Pcos(60o) and Px = Psin(60o) or Py = Psin(30o) and Px=Pcos(30o) (This is because in this case the Vector will make 30o with horizontal)

Subsets

A collection of things which have something in common as per the rule is called a set. For instance set of colors would be {red, green, yellow, white, blue} this set represents the collection of five different colors, these are called the elements of the set.

The elements are separated using commas (,) and every element is unique in the set. The notation uses parenthesis which are curly or in other words the flower brackets {} and usually a set is denoted using a capital letter.  Let ‘E’ be a set denoting all the even numbers less than ten; so set E={2,4,6,8}. While learning about sets we come across finite set, infinite set, universal set, empty set, sub-set and power set.

In this article we shall learn more about a Sub-set. Consider a set A={1,2,3,4,5,6} and set B={2,4,6}. When we compare both the sets it is clear that all the elements of set B are present in the set A and hence we can call set B as the sub-set of set A.

Definition of a Subset can be given as, a set B is a sub-set of the set A only if every element of set B is in the set A. The subset sign is ‘⊆’.

For instance, set A={orange, pineapple, apple, grapes, kiwi, mango} and set B={apple, kiwi, orange, mango}. Here each element of set B is in the set A and hence Set B is a sub-set of Set A and is denoted as B⊆A. Suppose Set P={2,3,4,5,8} and set Q={1,3,7,9}; each of elements of set Q are not in the set P and hence we cannot call set Q a sub-set of set P.
So, subset meaning is the set which is a part of the whole set.

So, a subset in math can be better explained using the following example, a set P is a sub-set of set Q if and only if all the objects or elements of set P is in set Q.

If set P={x, y, z} and set Q={x, y, z, p, q, r}, every element of P is in Q and hence P⊆ Q. Some of the subsets examples are, list of all the sub-sets of the set A={a, b, c} can be given as { },{a},{b},{c}, {a,b},{a,c},{b,c}, {a,b,c}.

The number of sub-sets of a given set can be given by the formula 2^[n(S)] where [n(S)] is the number of elements in the set. If there are three elements, the number of sub-sets is 2^3 which is 8 sub-sets in all.