Parallelogram Vector addition

One of the most common vector operations that is frequently encountered is addition operation. Addition of two or more vectors to arrive to a vector sum is called vector addition. Consider two vectors,  u=(u1, u2) and  v=( v1, v2), the sum of these vectors would be vector u + vector v= (u1+v1, u2+v2). This sum is called the resultant vector. There are various methods to find the resultant vector namely, parallelogram method, component method, graphical method, cosine method, polygon method etc.  

Parallelogram Vector addition:  In this method first two vectors are drawn such that their initial points coincide. Then the other two lines are drawn to form a parallelogram. The resultant would be the diagonal of the parallelogram drawn from the initial point to the opposite vertex of the parallelogram.

Vector addition component method is one way used in adding vectors. Component means ‘part ’and hence they can be considered as the coordinates of the point that is associated with the vector. In a Euclidean plane consider two vectors, u=(u1, u2) and v=( v1, v2), the resultant vector which is the sum of these vectors is given by, u+v = (u1+v1, u2+v2). In a three dimensional space, given vectors u=(u1, u2, u3) and v=( v1, v2,v3) the method would be similar to the method used in addition of vectors in a Euclidean plane. So, u+v = (u1+v1, u2+v2, u3+v3). We can finally conclude that vector-addition is just like the normal addition, component by component.

Let us now learn the vector addition graphical method, consider two vectors, u=(4,3) and v=(1,4) in the plane. Using the component method of vector-addition the sum can be given as, u+v = (4+1, 3+4) = (5,7). Using the graphical method we get the same resultant vector by taking one vector whose direction and magnitude is unchanged and placing its end at the unchanged vector’s tip, and joining the origin and the new location of the displaced vector using an arrow. This procedure in general works for addition of vectors. For any two given vectors u and v in the plane, the sum of the vectors in general can be graphically represented as the vector addition diagram given below

As the vectors in a two dimensional space lie in the same plane, any two vectors in a three dimensional space also lie in the same plane and hence graphical method works well for vector-addition in a 3-dimensional space.

Understanding the Concept of Linear Combination

In mathematics the equations are very important. The problems can be converted into equations and then the equations can be solved with the help of multiple methods. The solutions obtained must be checked for their feasibility. This is because all the solutions obtained will not be feasible. Only the solutions that are feasible must be selected, otherwise the answer might go wrong. One must be very careful in selecting the solutions of the equations.

In case of linearity the degree of the given equation is ‘1’. The linear combination can be formed with the help of an equation. There need to be constants for this process to be performed. It also involves the simple process of addition. Basically it is the formation of an expression. This expression can be formed with the help of constants and the simple addition process.

The term in the expression must be multiplied with a constant and the answers obtained must be added. This will give the required solution. The linear combinations can be very helpful and can have various applications in mathematics.

The process of solving linear combinations must be learnt thoroughly to appreciate the concept. The examples will help in better explaining the concept. This concept is related to the concept of linear algebra. It is a branch of algebra and has to be learnt to understand this concept. This concept is also very helpful in vector theory and is a part of vector theory.

Vectors are different from scalar quantities. Both are quantities which have magnitude. The difference between vector and scalar quantities is that the vector quantities have a direction attached to them. But the scalar quantities do not have a direction attached to them.

This is the basic difference between the two quantities. So, one has to be very careful in dealing with these types of quantities. The direction can play a very important role. If the direction is not properly denoted the answer can go wrong. So, in the case of vector quantities, the direction plays a very important role. But in case of scalar quantities the magnitude plays the most important role.

The change in magnitude can make the whole answer go wrong in the case of scalar quantities. But in the case of vector quantities both magnitude and direction are important. Even if one goes wrong the whole answer goes wrong.

Definition Subset

Sub-set means part of. If we say that America is part of world then America is said to be sub-set of world. So we Define Subsets as the set which is part of another set. For example in library are many book shelves according to the subjects.
Consider a book shelf containing English books. Then the English book shelf will be sub-set of all library shelves.
Consider students in your class which makes a set C and all students in your school which makes a set S. the set C will be sub-set of set S as it is a part of set S or we can say it is contained in set S.  :

Definition Subset / Subsets Definition – sub-sets are those sets whose all elements are contained in some another set.

Let us see properties of sbsets:
Symbol used for sub-set is:  or . Example if A is sub-set of B then we write it as: A B or A B.
A sub-set may be sub-set of more than one set. For example: consider set N = {11, 33, 77}. This set can be sbset of any set containing the elements 11, 33 and 77 like {1, 2, 11, 33, 77}, {a, e, 33, 11, 77} and many more.
All sets are sub-set of themselves. For example set S={English, math, science, GK}. Then the set S contains these elements. We can say that S is sub-set of S.

An empty set (φ) is sub-set of itself and also sub-set of all other sets.
A set is said to be proper sub-set of another set if another set contains the sub-set and also one or more extra elements with it. For example: set A = {c, v, r, e, m} is proper sub-set of B = {c, v, r, e, m ,s} as it contains extra elements s with the elements of A. A set can be proper sub-set of more than one set. For example above set A is also proper sub-set of C ={c, v, r, e, m, d, s, a}. Null set is proper sub-sets of all the existing sets.

If A is sub-set of set B then B is known as super set of B.

A superset can also be sub-set of some other set.

If A is sub-set of B and B is also sub-set of A, then A is equivalent set of B or A=B.

Possible Number Subsets in set A = 2n where n is number of elements in set A.
These are the properties of Subsets Math

Venn Diagram Math

The relationship between two or three sets can be shown diagrammatically in the form of circles called the Venn-diagram. They were first given by the mathematician John Venn and hence the name for the diagrams. As we know set is the collection of unique things which are called the elements of the set. Consider the following Venn Diagram examples, List of odd numbers less than 15 forms a set which can be denoted as set A={1,3,5,7,9,11,13} here each element is unique.

Let us now learn how to draw Venn-diagrams to show the relationship between two sets. Let set A={1,3,5,7,9,11}, a set of odd numbers and set B={1,2,3,5,7,11} set of prime numbers.
The Venn-diagram of these two sets would be as given below,

In the above diagram the two circles represent each set, set A and set B. Both the circles with all the numbers represent ‘Union of sets’.

The green shaded region represents the numbers in both the sets, this is called the ‘Intersection of sets’, the yellow region represents only A, that is the element present in only set A and the orange region represents the element present in only set B.

Thus one glance at the Venn-diagram gives all the possible information which gives a logical relationship between the sets. At times we see Venn diagram lines, these lines are provided to write the elements or at times lines are used to show the relationship between the sets using line shading.

Venn diagram can also be used to show the relationship between three sets which would be a three circle Venn diagram.

For example: In a group of 23 students, 12 read mathematics, 15 read statistics and 11 read Physics, 4 read Mathematics only, 7 read Statistics only, 3 read Physics only, 4 read all the three subjects, 1 reads only Math and Statistics not Physics, 3 read only Mathematics and Physics not Statistics and 3 read only Statistics and Physics not Mathematics. Let us now write down all the given data to draw the 3 circle Venn diagram.

The total number of students is denoted using the symbol ‘µ’ which is ‘mu’, n(MUPUS)= µ=23, Mathematics =n(M), Physics=n(P) and Statistics=n(S). Intersection is shown with a symbol ‘∩ ‘showing the elements common to both the sets.

Elements not present in a set is shown by compliment, not A is A’. n(M)=12, n(P)=11 and n(S)=15. n(M∩P∩S)=4 and hence, n(M∩P)=7, n(P∩S)=7 and n(M∩S)=5. In the Venn-diagram below each color represents a logical relationship,

Least Common Multiple of two Numbers

There are various mathematical concepts and terms that are to be learnt. The term LCM stands for least common multiple. This is an operation and can be carried out on two integers. The answer we obtain is the least common multiple. This denotes the least number that is fully divisible by both the integers.

Now how to find LCM is the question. It is a rather an easy task and can be performed if one is familiar with the basic arithmetic operations of mathematics and number theory. Division is the operation that is most frequently used. So, one must be confident in the division of two numbers.  So, to understand how to find the least common multiple one must be thorough be with the basic concepts in mathematics otherwise it would be really tough to proceed ahead.

So, what is LCM and how can be it found out. The numbers are selected first, for which it has to be found out. Then they are divided by a common number which should be able to divide all the numbers chosen completely. The number which is not divisible by the common number is written as it is for the next step to be performed. In the next again the numbers are divided by a number which is able to divide all the chosen numbers and this continues till ‘1’ is obtained at the end of all the steps of division. The numbers chosen are divided several times till ‘1’ is obtained in the end. Then the product of all the numbers which were used for division is obtained and this gives the least common multiple of the chosen numbers in the beginning.

So, what is the LCM of ‘3’ and ‘2’ is a question that must be answered and can explain the concept better. First a number which divides both ‘2’ and ‘3’ have to be found out. Since there is no common number that divides both ‘2’ and ‘3’ their product is found out. On multiplying ‘2’ and ‘3’, ‘6’ is obtained. So, ‘6’ is the least common multiple of ‘2’ and ‘3’. If two numbers are not divisible by a common number in the beginning itself, then their product itself is the least common multiple of the two. This is apt in the case of two prime numbers. In the example both ‘2’ and ‘3’ are prime numbers.

Multiplying Even and Odd Numbers

Even and odd numbers are two of the most important topics in basic arithmetic. All real numbers can be classified into even numbers and odd numbers. Even number is a number that can be evenly divided into two parts or can be exactly divided by 2. For example: I have to shed almost 12 kg post pregnancy weight . Here, 12 is an even number. On the other hand, odd number is a number that cannot be evenly divided into two parts or cannot be exactly divided by 2. For example: I have already shed 7 kg post pregnancy weight . Here, 7 is an odd number. There are some trend that can be seen while multiplying even and odd numbers. Let’s have a look at the multiplication rules for even and odd numbers in this post.

Multiplying Even Numbers
When two even numbers are multiplied, the result is always an even number. The formula is Even * Even = Even. For example: I need to list 6 different hobbies for kids for 2 magazines. 6 hobbies for kids * 2 magazines = 12. The result is an even number.

When an even number is multiplied by an odd number, the result is always even. The formula is Even * Odd = Even. For example: I have to survey minimum of 10 baby care center across 5 cities. 10 baby care center * 6 cities = 60. The result is an even number

Multiplying Odd Numbers
When two odd numbers are multiplied, the result is always an odd number. The formula is Odd * Odd = Odd. For example: I have to buy 7 play toys for 7 kids. 7 * 7 = 49. The result is an odd number.
When an odd number is multiplied by an even number, the result is always even. The formula is Odd * Even = Even. For example: She has to buy 7 dresses for 4 kids. 7 * 4 = 32. The result is an even number.
These are the multiplication rules for even and odd numbers.

Outer product of vectors in R^3

Outer product of vectors in R^3
Cross product definition:
The outer product is also called the cross product of vectors. The formal definition is as follows:
If vector x = (x1,x2,x3) and vector y = (y1,y2,y3) are vectors in R^3, then their outer product is denoted by x X y and is defined as
x X y = (x1,x2,x3) X (y1,y2,y3)
         = (x2y3 – x3y2, -(x1y3 – x3y1), x1y2 – x2y1) <- cross="" is="" p="" product="" the="" this="" vector.="">Note that the cross product of two vectors is always another vector. Alternatively in matrix notation we can write the cross product identities as follows:
x X y = (|(x2&x3@y2&y3)|, - |(x1&x3@y1&73)|, |(x1&x2@y1&y2)|)

Properties of cross product:
(1) x X y = -y X x
That is because when we do y X x we interchange the rows of the cross product determinants. Interchange of rows results in negative value of the same determinant.

(2) x X x = 0
That is because if a determinant has two identical rows, then the value of the determinant is zero. When finding x X x, the two rows would be identical.

(3) x X ky = kx X y = k(x X y).
Here, k is a scalar. If one of the rows of a determinant has a common factor k, then the k can be taken out of the determinant and the resulting determinant value when multiplied by k gives us the value of the original determinant.

(4) x X (y + z) = x X y + x X z
This is just distributive property. The distributive property is applicable to determinants there for it also applies to the outer product.

Cross product rules:
The following rules apply to all outer products of vectors. The outer product is also called vector product of two vectors.

(1) The vector product of two vectors is always another vector.
(2) The vector product is not commutative as we saw in the first property above.
(3) The vector product is not defined for vectors in R or R^2. It is defined only for vectors in R^3 or higher order dimensions.

Sample problem:
Find the outer product of the vectors x = (1,2,3) and y = (-1,3,5).
Solution:
x X y = (|(2&3@3&5)|, - |(1&3@-1&5)|, |(1&2@-1&3)|) = (10-9, -(5+3), (3+2)) = (1,-8,5) <- answer.="" p="">