Thursday, 28 January 2021

PURPOSE OF DETERMINANT OF A MATRIX IN REAL DAILY LIFE .

 

SO WHAT IS THE PURPOSE OF THE DETERMINANT OF A MATRIX IN REAL DAILY LIFE ? 

A Determinant is a measure of how many times larger is the area (in case of a 2 x 2 matrix) or Volume ( in case of a 3 x 3 matrix) w.r.t the area or Volume a unit cube.

So Determinant = RATIO OF how much is the area of the object matrix under consideration divided by the area of a unit square.


 

For 3 dimensional figures (3 x 3 ) matrices, instead of area ------> substitute volume. And instead of unit square ---------> substitute area.

BUT FIRST, I WANT YOU TO KNOW ABOUT HOW MATRICES ARE MAPPED ONTO A GRAPH.



 


 


 

 

Every element of the matrix may represent a certain co-ordinate that the matrix is stuck upon or hung upon.

For example, just look at the matrix above.What you will see is that the number 3 in the left-most and top-most section of the matrix represents 3 on the x-axis. 


3 lies on the x -axis and there is no y and z component to it. This means that there is one point or one vertex of the matrix such that it has co-ordinates of (3,0,0) on a 3-D graph.

 KEEP SCROLLING BELOW FURTHER THIS PAGE TO READ ABOUT NEGATIVE VALUES OF DETERMINANTS .......






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book

 

  

 

 

 






 

 

 YOUTUBE VIDEO

 
 
 
 
 
APPLICATION OF MATRICES IN REAL LIFE VIDEO


RELATED LINKS 

RELATIONSHIP BETWEEN DETERMINANT AND VOLUME OF A MATRIX


Determinants of a matrices give a relative volume of a matrix as compared to the volume of a  unit matrix. 

This unit matrix will be a unit square if you are dealing with say a 2 dimensional matrix.

This unit matrix will be a unit cube if you are dealing with say a 3 dimensional matrix.

WHAT IF THE DETERMINANT OF A MATRIX IS -VE ? 

Say you get the determinant of a matrix as -2 .

It still means that the volume of your matrix or the area that your matrix is trying to denote is 2 times that of the unit cube or unit square.

THE ABSOLUTE VALUE OF YOUR MATRIX IS WHAT COUNTS
  
Not whether it is positive or negative.


 


Saturday, 23 January 2021

INVERSE OF A MATRIX WITH APPLICATION AND PURPOSE IN REAL DAILY LIFE






 


Analogy / Example for inverse of matrix

Imagine a software through which if you put an image(digital), this image stretches in width by a factor of 4

and its height increases by a factor of 3/2 .

 

Now lets say you want to make a software that has to bring the image back to its original dimensions.

This software does this by multiplying the stretched and increased height modified matrix by a new matrix such that the resulting matrix will

be the original unstretched and unmodified image.

 

The parameters of the unmodified image can be represented as Matrix x [ Identity Matrix] .

Just as every in numbers, every value can be represented as number x 1,  For example your weight can be represented as 65 x 1.

In a similar manner in the world of matrices, every matrix can be represented as Matrix x (Identity matrix).

The number 1 when multiplied to any number means an operation which does not change anything about the number. Its actually multiplication

by the fraction 1/1. We just loosely write it as the number 1. Its actually a fraction.

 

Now imagine you are going in a rocket and suddenly your rockets speed increases by 4 times. And you need some mechanism to slow the rocket down

to how it was before the power surge happened. You would require some software that will reduce the speed of the rocket by 1/4.

As you see, 1/4 is the inverse of 4.

 

Now lets say its a car that has dented due to an accident. Certain dimensions of the car have dented inwards by a factor of 1/1.2 

The inverse of 1/1.2 is 1.2

This entire incident of denting and undenting can be represented by matrix multiplications. That matrix which will produce  the undenting and

recover the original dimensions of the car is known as the inverse matrix.






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book

 

  

 

 

 






 

 

Thursday, 14 January 2021

VISUALIZING FLUX AND FLUX EQUATION INTUITIVELY.



EXAMPLE OF FLUX

Flux is any effect or influence like heat or magnetic field or electric field which usually has the ability to pass through substances and surfaces.

Flux describes how much of an influence is passing out through a surface. 

Eg:- X - Rays,

 

ANALOGY FOR FLUX

Hair growing from your scalp or grasses growing from the ground. The denser they are, the more you speak of flux density.

 

PARALLEL AND PERPENDICULAR COMPONENTS OF FLUX

 

 

Flux has a component perpendicular to the surface. And it has another component parallel to the surface. Only the component perpendicular to the surface has a significant effect . This component perpendicular to the surface will be parallel to the normal 'n'. The normal is a line drawn perpendicular to the surface.

So the line that is perpendicular to the surface will be parallel to the normal. Because the normal itself if an imaginary line perpendicular to the surface. So any flux line perpendicular to the surface will be parallel to the normal and these are the only lines which have any significant

So another way of putting it is that only that flux components which is parallel to the normal ( and  thus perpendicular to the surface ) will have any significant effect .

 

EQUATION OF FLUX

It is  Flux strength experienced = How strong the field intensity is  x How large the surface area is of the surface through which flux is passing x how parallel the normal (an imaginary perpendicular line drawn to the surface is ) x how parallel the field lines are w.r.t the normal drawn to the surface (this angle is represented by cos(thetha)

The above explanation in formula format is as.....

F = E.A.cos(angle)

 

FLUX IS ZERO WHEN FIELD IS PERPENDICULAR TO THE SURFACE ( And thus parallel to the normal).

ANALOGY FOR THIS IS AS FOLLOWS

Now imagine that this arrow is being shot northwards. It will hit the target because the target seems to face it.

But now imagine an arrow being shot from west to east or vice-versa. Will this east-west arrow hit the target at all. No, right ? Because it runs parallel to the plane of target (bullseye plank ).

 

COS( ) COMES INTO MATH WHEN 2 FORCES/INFLUENCES ARE SUCH THAT THEY PRODUCE MAXIMUM IMPACT WHEN KEPT PARALLEL TO EACH OTHER.

So, if in the equation , you see cos( ), that means that net forces will be maximum when the influences are parallel to each other.


 

 

 

 

 

 


SOURCES

WHY ELECTRIC FLUX IS ZERO IF ELECTRIC FIELD IS PERPENDICULAR TO SURFACE

 






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book

 

  

 

 

 






 

Tuesday, 29 December 2020

APPLICATIONS OF COMPLEX NUMBERS OR IMAGINARY NUMBERS IN ELECTRONICS

                   Whenever a force/influence etc gets split into two daughter forces/influences or two branches,...complex numbers come into the picture. 

In electronics, when you put a resistor in the path of electrons, all that they do is burn electrons. So there is only one effect the overall circuit has on the army of electrons, burning.

But now when you put a capacitor or an inductor in the path of the incoming electron army, what do they do ?

 They delay the flow of electrons. They as if act as a traffic police or say a dam. They hold electrons for some time and then they leave these electrons .

So now there are two daughter branches of effects on the incoming army of electrons. 

1) The resistance which fries electrons .

This is also called as the real part.

2) The capacitance which just delays the electrons.

This is also called as the imaginary part.

Yes, I too am searching for the guy who came up with the nomenclature of real and imaginary parts . One day I will get him in my hands.

 






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book

 

  

 

 

 





Friday, 20 October 2017

COMPLEX NUMBER EXPLAINED SIMPLY

complex numbers simply explained with visual image
|||| The traditional number line is capable of denoting or representing only forward and backward motion. The numbers that lie on the number line such are called as REAL NUMBERS.


|||| But what if an object is moving in an inclined plane like a sphere. Now there is an added dimension to it,...right. The number line by itself cannot fully represent it. It needs help.That is where the complex plane comes into picture.






|||| The REAL PART + COMPLEX PART now  together, describe the complete motion of the object.

( )  The real component is the component which is usually the one parallel to the x-axis.
( )  The imaginary component is the one which is usually the one perpendicular to the x-axis (and parallel to the y-axis).

||||  However, even when a number like 5 gets split up into 3 + 4i when being represented on the complex plane,....the magnitude remains the same as sqrt ( 3^2 + 4^2) = 5. 

||||  The numbers that can be handled by the number-line are called as REAL-NUMBERS.


||||  The numbers that cannot be handled by the number-line are called as COMPLEX-NUMBERS.



APPLICATION OF COMPLEX NUMBERS OR IMAGINARY NUMBERS IN ELECTRONICS


THE FAMILY OF REAL NUMBERS
||||  The family of REAL-NUMBERS include all the numbers we learn about in high school like the 
Rational Numbers :- Those numbers not having a square root with them.
Integers :- Positive or Negative but those numbers not having a decimal point or any fraction involved with themselves.eg -2, -7, 8 , 5 , 2
Fractions :- You know what they are. 
Irrational Numbers :- Those numbers which have a square-root attached to them. So why call them Irrational number:- I pondered about it and jokingly I came up with a reason. Maybe it would be irrational on our part to say that the car is moving at sqrt(3600) km/hr....or that my marks are sqrt ( 1600 ).

FAMILY OF COMPLEX NUMBERS.

||||  Ok, I dont know specifically of any types but i do know that i , j and k maybe used to denote the three different types in which inclination can happen.



So +i and -i will denote anticlockwise and clockwise motion repectively.
And +j and -j will denote rotations in a horizontal plane. 
And +k and -k will denote rotations in a vertical plane but this time back to front or front to back. 

NOW GENERALLY SPEAKING ABOUT COMPLEX NUMBERS...
|||| Complex numbers are used to denote inclination.
|||| Yeah....thats it.

|||| ‘ i ‘ as if denotes inclination. If there is no i, that means the object is completely MOVING JUST FORWARD OR BACKWARD....along a thread or a rope or a flat surface but in a line.


|||| So if the complex part is completely zero,only the real part remains....which means that there is no inclination of any kind. The object now is moving in a straight line (maybe even parallel to the x-axis many of the times).












||||  Now, one could also say that Complex numbers are used to denote rotation. But then inclination and rotation are both the same.
||||  You cannot incline something without rotating it and you cannot rotate something without inclining it.

WHAT DOES 3+4i MEAN?
















|||||| i denotes that the object is placed at an inclination.
|||| 3 is the ‘horizontal influence’ of that object. And 4 is the ‘vertical influence’ of that object.

|||| So, what happens is, when an object gets inclined, it’s influence gets divided into two parts (components).
1) A horizontal component
2) A vertical component


|||| The vertical component is sin and the horizontal component is cos.

WHERE ALL DO COMPLEX NUMBERS COME IN REAL DAILY LIFE ? AND IN OUR EQUATIONS?





|||| Complex numbers come into our lives when one force gets divided into two branches of forces.

|||| Inclining the object is just one example where the force will get divided into two.

EXAMPLE 
|||| Imagine a stick.



CASE 1: THE STICK IS COMPLETELY HORIZONTAL 


















CASE 2: THE STICK IS SLANTING













CASE 3: THE STICK IS COMPLETELY VERTICAL


















|||| Whenever one force gets divided to two components, its due to the force getting inclined or some other reason, the real and the imaginary parts come along. 







|||| The real part is the part which is usually parallel to the horizontal or to the base surface taken as a reference. For example in the above example, the ground is the base OR reference surface. 







|||| So, in this case, the real part is the part parallel to the ground surface (the shadow below). The imaginary part is the part perpendicular to the ground surface (the shadow on the wall).





|||| The real part is the horizontal influence (horizontal projection ). 





|||| Imaginary part is the vertical influence (vertical projection).



USE OF COMPLEX NUMBERS TO INDICATE ROTATION.
























































A VIDEO ON THIS.

AN ALTERNATE VIEW OF COMPLEX NUMBERS.GIVEN BELOW. 
                               LINK GIVEN HERE YET ANOTHER WAY TO VIEW COMPLEX NUMBERS
A DIFFERENCE IN PERSPECTIVE AS GIVEN ON BETTEREXPLAINED.COM
|||| The above table given on better-explained.com give us a valuable insight on what was wrong with our traditional way of looking at things. Whenever we thought of complex numbers, we first imagined a number line, then we imagined taking square-root of a negative number, and going still backwards at the number line.

|||| However, complex numbers are all about revolving around the number line. That is the reason why the numbers 1, i , -1 and -i keep repeating cyclically in complex numbers. Want an example? i^0=1 i^1=i i^2=-1 i^3=-i i^4=1 i^5=i ............and so on. 


|||| This kind of cyclical behaviour is seen only when things move round and round about a same location usually. They call it cyclical for a reason.

|||| So this was the main problem with our imagination. We were seeing backwards whereas we were dealing with inclination and rotation and a cylical activity. 
|||| You can compare 1,i,-1 and -i to the EAST-NORTH-WEST and SOUTH Poles in the same order in which they are written. 
|||| If you stand facing the East and rotate 90 degree anticlockwise, you start facing North, you rotate another time by 90 degree from there, you start facing the west, and another 90 degree turn will make you face south and then again if you turn, you face east again.
       
A USEFUL VIDEO FROM BETTEREXPLAINED.COM

COMPLEX NUMBERS SIMPLY EXPLAINED WITH PURPOSE AND APPLICATIONS.

|||| Complex number indicate inclination almost always.
|||| Example 1
Imagine you opened a picture in microsoft paint. 
Now lets say you rotated that image by 90 degrees. That is "i".
i = inclination by 90 degrees.
i * i = inclination by 180 degrees.
i*i*i = inclination by 270 degrees.
i*i*i*i= inclination by 360 degrees.
                              and so on.

Why is complex numbers a part of maths?
A normal number line can be used to denote an increase or a decrease in length or distance or area etc.
||||However, can such a line help you when the object has just been rotated or inclined at an angle?
|||| This where the complex plane comes into picture. It is used to measure how much you are inclining the object.

WHAT ABOUT NUMBERS LIKE 3+4i?
|||| Is it necessary that an object should be inclined by 90 degree or 180 degree or 270 degree sharp?
|||| Cant it be inclined at a lesser or an intermediate angle.
|||| If you take tan inverse of 3 + 4i, you will get the angle by which it is inclined.
|||| 3 is the view you will get if you stands upon the x-axis and look upwards towards the object (i.e the top view)  
|||| 4 is the view one will get if climb up the Y-axis like climbing up a pole and then try to view that object (i.e 4 is the front view of the object)
|||| Now what is the actual length of the object? 
|||| It can be found by sqrt(3*3 + 4 *4).

WHAT DOES THE COMPLEX PLANE LOOK LIKE?
|||| In many ways it looks like a map having NORTH, SOUTH and EAST, WEST arrows.
|||| The only difference : Instead of North-South, there is i and -i.
|||| And instead of East-West, there is 1 and -1.
|||| Now let's say that you are standing facing towards the East. This is like going towards i on a complex plane and so on.
|||| Comparing these two planes, we can say the following,
 East ~ 1
 North~ i
 West ~ -1
 South ~ -i.

THE SIGNIFICANCE OF THE NUMBER 1 IN MATHS.
|||| 1 is a number which denotes 'full' or 'complete' of anything.
|||| Just a 1/2 denotes 'half' of the full, 1/3 denotes 'one-third' of  the full, 1/4 denotes a 'quarter' of the full,......1 denotes 'full itself'.    

  
|||| Similarly -1 denotes 'full' by inverted upside down. Imagine a machine in which objects when put come out with the original shape and size, but made to point in the opposite direction. '-1' in maths is another way to say, 'Size,Shape kept as it is, but direction inverted by 180 degree.

NOW WHAT ABOUT 'i'?
|||| Instead of seeing it as i, you may see it as 1i.
|||| i is a mathematical way of saying, "Size, shape, kept intact, but rotated by an angle of 90 degrees.
|||| - i is a mathematical way of saying, "Size,shape, kept intact, but rotated by an angle of 270 degrees.

COMPLEX CONJUGATES.
|||| 3+4i  is like an arrow pointing 3 degrees eastwards and 4 degrees Northwards.
|||| 3-4i is like an arrow pointing 3 degree eastwards, but now 4 degrees Southwards.
|||| To do this, you just have to rotate the first arrow by 90 degrees vertically downwards.
|||| So thus, a complex conjugate is nothing but an object rotated by 90 degree vertically (upwards or downwards).

WHAT IS THE SIGNIFICANCE OF THE i, j and the k planes?
|||| To understand this, all you have to do is ask yourself, "In how many different directions(planes actually) can I rotate any object in my hand?
|||| You will see that there are 3 planes in which this can be done.
1] You can rotate the object in a x-plane, or in a y-plane or a z-plane.
2] Just as we have x,y and z co-ordinates to pinpoint the magnitude of an object or force, we have the i,j and k co-ordinates in maths to pinpoint the direction of rotation of the object.

COMPLEX NUMBERS AND ITS TRUE SIGNIFICANCE AND APPLICATIONS.
|||| Complex numbers come in mathematics when a force gets divided into two branches due to its inclination. Want a example?
|||| Imagine a missile which strikes a ground initially at 90 degree. Let's say its force was 5 kN.
|||| Now imagine another case in which the missile hit the ground inclined manner.  Now the force of the missile gets branched into two components as follows
1] A vertical component which is calculated by 5sin(angle of incidence).
2] A horizontal component which is calculated by 5cos(angle of incidence).
So whenever a Full force gets divided into two or more forces due to rotation or inclination or such other phenomenon, complex numbers come into picture.
|||| To picturize the significance of complex numbers, just imagine a scenerio where complex numbers are not there. Instead of writing the component forces as 3+4i, say we write it as 3+4, now wont the reader feel that these two numbers have to be added and the result should be 7?
|||| That is where complex numbers come in. They warn us that the 3 and the 4 are not meant to be added together as we did in high school. That things are a bit, hmm what to say 'complex' in this case?
|||| One needs to understand that the net force of 5 got divided into two branches , a vertical branch of 4 and a horizontal branch of 3 due to an inclination of an angle of tan inverse of (4/3)....(whatever the answer....sorry, am a bit lazy to go to the calculator right now).

SOME USEFUL LINKS

A VISUAL GUIDE TO UNDERSTAND COMPEX NUMBERS.

WHY COMPLEX MULTIPLICATION WORKS 

COMPLEX NUMBERS VIDEO 1

COMPLEX NUMBERS VIDEO 2 

COMPLEX NUMBERS SOMEWHAT AN OK VIDEO

|||||||||||||||||||||||||||||||||||||||||||||| 
|||||||||||||||||||||||||||||||||||||||||||||||||||

BUY

THE ABOVE IS A EXCERPT FROM A BOOK AVAILABLE FOR PURCHASE (5$) AT THE FOLLOWING LINK
AVAILABLE FOR PURCHASE FOR 5$ (INR 300) HERE (PDF)

||||||||||||||||||||||||||||||||||||||||||||||

DONATION SECTION












DONATIONS SECTION HERE

||||||||||||||||||||||||||||||||||||||||||||||||||



ABOUT
I hardly understood Math in school. In fact was on the verge of dropping the subject I loved the most because as much as I loved the theory of it, I could not understand the math involved in it. However I loved the subject too much to be able to live without. Hopelessly, I was continuing my love-affair with it. 
Then one day....a miracle happened,....while applying a certain formula again and again.....I came to know its significance. Slowly and steadily....other equations also started clicking. I got to see a strong relationship between Maths and the Physics it was pointing towards. They both were the same. Maths was just an easy language to express a physical phenomenon. And a bit more to that in the sense that it could even predict the behaviour of a certain physical phenomenon. Equations now as if came to life. Every equation now had as if something to say. A burning urge to share these things with the world aflamed within me. So that no one has to give up the subject that he or she loves the most. 
The book on visualizing maths thus got written as a sprout of inspiration. The blog followed. Both these are dedicated to you and all such similar minds searching for answers.
Visualizing Math & Physics is a blog dedicated to you and all such similar minds searching for answers.
||||||||||||||||||||||||||||||||||||||||||||||
CONTACT binnoypanicker@gmail.com
||||||||||||||||||||||||||||||||||||||||||||||
"zone name","placement name","placement id","code (direct link)" visualizingmathsandphysics.blogspot.com,Popunder_1,15906098,""