WHY IS SIN(30) = 1/2 & NOT SIN(45) = 1/2 ??

( ) It was much later I understood why is that values are a bit out of place. 

( ) Carefully consider sin (45) .

( ) Sin(45) has a 1/2 in it but with a square-root. Sin (45) = sqrt (1/2).

So now why does this square root come? Good question. Thank you.

( ) Squares and Square-roots come in Math whenever two forces/events occur simultaneously to produce a net effect.

( ) The two forces or events will be of equal magnitude obviously.

( ) If the two forces produce a net increase.....squares come into the picture.

( ) If they produce a net reduction....then square-roots come into the picture. 

( ) So lets say you tilt a rod which is producing a shadow at a wall. At 90 degrees it was producing full shadow of itself upon the wall.

( ) But now you catch its one end and decide to tilt it to a lower angle say 60 degrees. Now when it tilts to a lesser angle.....both its ends are participating....its upper end moves towards the right say and its lower end moves towards the left say.

( ) These two events are of equal magnitude....and are happening simultaneoulsy and are producing a net reduction in the shadow projected upon the wall. So square roots come into the picture.

( ) Also note if instead of two it were three events contributing to the net decrease....then that is when cube-roots come into the picture. If the three events contribute to the net increase....then only cube.
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THE ABOVE IS A EXCERPT FROM A BOOK AVAILABLE FOR PURCHASE (5$) AT THE FOLLOWING LINK
VISUALIZING MATHS.PDF

AVAILABLE FOR PURCHASE FOR 5$ (INR 300) HERE (PDF)

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DONATION SECTION

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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.

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CONTACT
binnoypanicker@gmail.com

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VECTORS: Significance &Application of Cross product and Dot product.

DOT PRODUCT.

|||| A.B = |A||B|Cos(angle).

EXAMPLE

Haven't you wondered why cos?

|||| Like I said in my post on TRIGONOMETRY, Cos comes into equations when two forces work hand in hand-in-hand with each other to produce the net resultant.

|||| Dot product is also similar. It is used when Two forces A and B produce maximum effect or impact when work together 'in a line'.

|||| A.B = |A||B| Cos(angle) means that the resulting force  is due to 

1] Force of A

2] Force of B

3] Angle in which A and B are colliding into each other.

|||| Cos in the equation is like telling that, "if you want maximum impact, place A are B as parallel to each other as possible".

|||| For example, if you are rowing a river, row as parallel as possible to the flow of the river to get maximum results.

|||| Multiplication again means combination. Thus A and B and the angle of  A and B working together as if are combining to form the net result.

CROSS PRODUCT
|||| Cross-product is a measure of how much perpendicular two objects work in relation to each other.

|||| Cross product comes into picture whenever two objects work against each other, completely out of line and produce maximum impact when perpendicular.

|||| Now imagine the same boat and the same river, but this time instead of measuring the velocity of the river, you want to measure the friction between the boat and the river .

|||| In other words, instead of trying to measure maximum velocity, you want to produce maximum friction between your boat and the river.

|||| Now this will happen when your boat is at 90 degrees to the flow of the river.

|||| Thus maximum friction is when your boat is at cross with the river.

|||| Thus A X B = |A||B| Sin (angle).

|||| Sin as I said comes into those equations where max impact is at perpedicular angles or when two things work against each other or out of alignment.

|||| Thus we can say that Cross-product comes when we have to measure those forces like friction which happen when two objects work against each other and Dot-product comes when we have to measure those forces like total additive velocity which happen when two forces work hand-in-hand with each other. 

Hope this helps 

BINNOY
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THE ABOVE IS A EXCERPT FROM A BOOK AVAILABLE FOR PURCHASE (5$) AT THE FOLLOWING LINK
VISUALIZING MATHS.PDF

AVAILABLE FOR PURCHASE FOR 5$ (INR 300) HERE (PDF)

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DONATION SECTION

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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

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USEFUL LINKS

A GOOD EXPLANATION FROM BETTER-EXPLAINED.COM

A GOOD EXAMPLE OF DOT PRODUCT
ANOTHER EXPLANATION OF DOT PRODUCT 




DOT-PRODUCT AND CROSS-PRODUCT LINK FROM PHYSICS CLASSROOMS.COM

DOT PRODUCT EXAMPLE IS GOOD

A VIDEO

WHAT DO SINE WAVE INDICATE, THEIR PURPOSE & EXAMPLES.

 

||(/\)||Sine-waves come in mathematics to represent two opposite motions or energies.....the ying and yang the night and day, the up and down so on....||(/\)|| The smoother a sine-wave it....that means the smoother the transition is from up to down or from night to day from summer to winter etc.||(/\)||Infact if you take the example of duration of days every year....that too will form a sine-wave if plotted on a graph.

||(/\)|| The duration of days will keep increasing and increasing smoothly and gradually until it reaches June 21.||(/\)||June 21 if I am not wrong is some Summer Solstice...the longest day of the year. This will form our peak of the sine-wave. ||(/\)|| From June 21st however the length of daytime will keep on reducing smoothly and gradually until it reaches Dec 21st.(Still scared the world will end?)||(/\)||December 21 is the shortest day of the year.This is going to be the bottom peak of our sine-wave.||(/\)|| From December 21st forward on however the length of days will increase and the process will keep repeating.THE WASHING MACHINE ANALOGY|( )|Water in a washing machine moves to and fro in a circular manner.|( )|If you plot this on a graph, what you get is a sine-wave.|( )|Lets say when the water moves towards the right you plot it upwards on the graph.|( )|Initially the water picks up speed in the right direction until it reaches the maximum speed. This is the peak of our sine wave. |( )|Then the water starts losing speed and comes to a halt. This is when you see the gradual descent from the peak. When the water is at halt...that is the zeroth location of the graph.|( )| After halting at the zeroth location...now the water starts moving in the left direction.|( )|It catches speed and until it reaches the peak speed in the left direction. This is the bottom peak of the sine-wave....and then the process repeats.
A.C & Sine-waves.|( )|Electricity called as A.C which is alternating current also moves much similar to water in the washing machine in alternating directions. |( )| Ever wondered why it is called as alternating current? It alternates its direction every now and then. It is more like a churning of  electricity through the conductor. 

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BUY

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

BUY FULL BOOK OF VISUALIZING MATHS (PDF)

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DONATION SECTION

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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

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SOME COOL LINKS ABOUT SINE WAVES.Amazing animations on sinewaves here.Another page of great animation for sinewaves hereAn intuitive understanding of sinewaves from betterexplained.com here. 
giphy.com is famous for gif animation such as..

Enjoy giphy.com here.
One interesting link on sine waves here

&
and here

& more on sine waves here

WHY THE SQUARE ROOT OF A NEGATIVE NUMBER IS A COMPLEX NUMBER?

|( )| Why is the square root of -4 = 2i ?

|( )| Why does  the square root of any negative number  involve an i term?

|( )| Well…..turns out that whenever you are even talking about taking the square root of a negative number, you are actually moving in a circular path (complex plane).

|( )| If you were moving  on a linear track or dealing with anything increasing in a linear fashion…..you would be dealing with numbers like 1,2,3,4 etc  or even -1,-2, -3 etc.

|( )| None of these numbers  when squared will give one the answer as -4.

|( )| Even a negative number like -2 when squared will not give -4.

|( )|  -2 x-2  will yield 4.

|( )| Only a number like 2i when squared will yield -4 as the anwer.

|( )| But then isn’t a number like 2i itself a hint that you are not moving in a linear path and that your path is complex (completely circular maybe).

|( )| Thus a number like square root of -4 makes sense only on the complex plane (circular of periodic path).

|( )| It doesn’t make sense on the linear path.

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BUY

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

BUY FULL BOOK OF VISUALIZING MATHS (PDF)

AVAILABLE FOR PURCHASE FOR 5$ (INR 300) HERE (PDF)

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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

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