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Three Dimensional Geometry (part - 2) , The Planes | study material for IIT JEE | concept booster, chapter highlights/

THREE DIMENSIONAL GEOMETRY (PART - 2), THE PLANES










THE PLANESPlane is the surface such that if any two points are taken on it, then the line joining the two points lies on it.
General equation of plane is given as -  ax + by + cz + d = 0 , where a ,b and c is not equal to zero.
           POINTS TO BE REMEMBERED 1). a, b and c are the directions ratios of the normal to the plane ax + by + cz + d = 0.

2). Equation of the yz - plane is x = 0

3). Equation of the zx - plane is y = 0

 4). Equation of the xy - plane is z = 0

5). Equation of the any plane parallel to the xy - plane is z = c. Similarly for Planes parallel to yz and zx is x = c and y = c
EQUATION OF THE PLANE IN NORMAL FORMVECTOR FORMIf n̂ be a unit vector normal to a given plane and d be the length of the perpendicular from the origin to the plane, then the equation of the plane is given by -                           r.n̂ = d CARTESIAN FORMIf l, m, n are the directions cosines of the normal to the plane and d is the perpendicular distan…

VENTURIMETER , understanding the basic concepts and it's derivation

Today we are going to talk about the concept of VENTURIMETER. This device is the application of the Bernoulli principle. This is a very important device which is based on the laws of the hydrodynamics.
This device measures the flow speed of incompressible fluid which are flowing through the pipe of different cross sectional area. This device is also called flow meter or venturi tube.
This device consists of a horizontal tube or pipe with having wider opening of cross sectional area A1 and a narrow neck of cross section A2 , this two regions of the horizontal pipe are connected to a manometer which contains a liquid of density σ . At the wider opening the velocity of fluid is v1 and pressure p1 and at the narrower opening the velocity of fluid is v2 and fluid pressure is p2. And the density of fluid which are flowing through the pipe is ρ. For reference see the diagram below:

Now we are going to deriving the formula for speed of fluid , which are flowing through the tube or pipe. DERIV…

The speed of EFFLUX and THE TORRICELLI'S LAW , Understanding the basic concepts and it's derivations.

So , today we are going to talk about the concept of TORRICELLI 'S LAW and speed of EFFLUX. This concept is the application of Bernoulli principle. This is a very interesting phenomenon of hydrodynamics, this law is given by a Italian scientist  Evanglista Torricelli's in 1643. He calculate the velocity of fluid which are flowing as jet from a hole of a tank.
Going further in the Article , we have to understand the meaning of efflux first.
Then , what is efflux?
If we talk about the efflux then efflux is nothing ,  but it is a simple flowing of fluid just like a jet from a very tiny hole ie. Orifice , and it is necessary that the cross sectional area of the orifice is very small as compared to the cross sectional area of the tank.

Many of us must have seen , the flowing of fluid through a hole of a tank like as a jet , and we have seen that this flowing fluid falls to the ground at some distance from the base o the tank. This distance from the base of the tank is called range…

TERMINAL VELOCITY , understanding the basic concepts and it's derivation.

So today in "laws of nature " we are going to talk about, the concept of terminal velocity. It is very important and interesting phenomenon of hydrodynamics.
Everyone must have seen that when a stone and any other objects , fell down in the pond or a well and any other water bodies, then the speed of falling of stone in water has been changed and start falling with a constant velocity.
So, the question is , why velocity of stone has changed and acquired a different velocity?
Then the answer to this question is explained below:

According to the Archimedes principle , if any object is immersed fully or partially in the fluid then that object experiences a buoyant force and this buoyant force is equal to the weight of fluid which is displaced by the object at the time of immersing. Every object displaced the same amount of volume as the volume of the object.
It means when a stone is dropped into the water then the buoyant force start acting in the opposite direction of falling…

BERNOULLI PRINCIPLE, understanding the basic concepts and it's mathematical derivation//

So we are going to talk about a very important and interesting concept of hydrodynamics that is the BERNOULLI PRINCIPLE.
Daniel Bernoulli a Swiss physicist developed this concept in 1738.
See how old this principle are but it's usefulness and importance has never been fade till now.
This principle plays a important role in finding the speed of efflux in Torricelli law,
It is used in venturimeter , in sprayer, flying of aeroplane, blowing off of roofs in Strom, flying of birds etc, all these phenomena are explained with the help of Bernoulli principle.
So everyone should understand the underlying concept of this principle. So read it carefully and keep patience.
This principle is strictly based on the conservation of energy. However, we are talking about the topic of hydrodynamics. Then which type of energy are involved in it. When Bernoulli was doing his research then he observed that, if any ideal fluid that is non viscous , incompressible, streamlined and irrotational, then at …

The equation of CONTINUITY, understanding the basic concepts and it's derivation.

So today we are going to talk about a very important and interesting concept of hydrodynamics  i.e equation of continuity.

Ask yourself that
What is the mean of continuity? And why are we using it here in hydrodynamics?
Continuity means anything which covers equal displacement in equal time interval.
In fluid mechanics, if V volume of liquid is entering in ∆t then same volume of liquid is exitting in ∆t also.
Fluid has so many properties, but the property by which it became unique, is the property of incompressible, it means fluid can't be compressed by any means,
Here fluid means liquid,
In hydrodynamics we consider a ideal fluid,  because mathematical modelling of real fluid is very complicated.
A ideal fluid is that fluid which is non viscous, incompressible and irrotational.
So it is observed that when a ideal fluid flows through the pipe of different cross sectional area, the volume of fluid which is entering is equal to the volume of fluid which is exit. If fluid is ideal t…

POISEULLIE'S LAW, Understanding the basic concepts and it's mathematical derivation//

So today in "laws of nature" we are going to talk about a very important and interesting concept of hydrodynamics , that is poiseullie's laws.
So going further it is necessary to understand the underlying concept of poiseullie's laws in very simplest  form.
In poiseullie's laws fluid motion should be streamline, through any pipe.
So let's starts:
We are starting the concept with a question that in what manner a viscous fluid flows through a cylindrical pipe?
In any cylindrical pipe when a viscous fluid flows then they follow a convex meniscus,but why? It is because, we know that a streamlined fluid has several stream of fluid.
Then some streams been near the wall of the pipe and some stream is far from the wall, In the cylindrical pipe the longest distance from the wall is the centre of the pipe.
And we know that the pipe is cylindrical then the shape of water which are flowing through it, must be in the shape of cylinder.
Then we can conclude that every flu…

The process of BIRTH and DEATH of STARS, explained.

Today in "laws of nature" we are going to talk about the some basic concepts of birth and death of stars.
Stars and black holes are the such massive objects in the universe, which have a very long age of about billions years or more.
This observable universe is full of massive objects, and humans is nothing if we compare it to the universe.
So today we are going to learn about the birth and death of stars only.
So let's starts:
Our earth is a part solar system and this solar system is the part of the milky way galaxy, and in our milky way galaxy there are several hundred billions of stars of all different ages, sizes and masses.
Sun is a star and it is the head of the solar system, and a star is nothing but a huge cluster of hot gases and plasma, which  continuously radiate energy in the form of light and X- rays.
So because sun is a star, it also radiate X-rays to some extent, but this X-rays brust became larger during solar flare.

A star is a factory of nuclear reacti…

Why clouds seems us darker than usual during raining???

Today in "Laws Of Nature" we are going to talk about this interesting phenomenon,that why does clouds looks darker during raining. Everyone must have seen the clouds at the time of raining,it's seems dark or in grey colour, but as the rain stopped, one or two hours later cloud again seems us white, so we will discuss here, today that what makes clouds darker during raining and white when it is not raining.
So let's starts
We are all know that how clouds are formed and what are clouds in reality?
We know that when waters boils, it turns into vapours and vapours is nothing but it is only the form of water in gaseous state, in reality water vapours is not a gas, but due to gaining high external energy it split into small tiny droplets of water and become unstable, and starts moving in random motion as particle of gases move. So sometimes water vapour is considered as gas.
This water vapour rises higher in the sky and gets condensed to form clouds, as it rises up, the t…

LCM and HCF, Problems tricks | fast track formulae for Problems solving.

In "Laws Of Nature" today we are going to talk about the some important arithmetic formulae on LCM and HCF, this formulae may be helpful to you, in all types of competitive exams.

LCM AND HCF PROBLEMS TRICKS
* To find the HCF of two numbers.
Method-
Split the given numbers into prime factors and then find the product of all the prime factors, which are common to all the numbers.the product is the required HCF.

* To find the HCF of decimals.
Method-
First make the same numbers of decimals places in all the numbers,and then see the numbers as integer and find the HCF, and in the answer put the decimal places as there in each of the numbers.

* To find the LCM of two numbers.
Method-
Split the numbers into prime factors and then find the product of the highest power of all the factors that occurred in given numbers.the product will be the required LCM.

* To find the LCM of the decimals.
Method-
First make sure the same numbers of decimals places in all the numbers.then find the …

After observing the earth from a helicopter, WHY DOES EARTH NOT SEEMS US ROTATING???

In "Laws Of Nature" today, we are going to talk about a famous natural phenomenon that if earth is rotating at the speed of 460m per second at the equator,then if we jump on the surface of the earth then why does earth not seems us rotating.
Or in a other examples if we observe the earth from a height i.e from a helicopter them why does earth not seems us rotating, these both examples are based on the same concept,and today this concept is going to revealed,this type of questions is very popular in students and it is a very interesting phenomenon.

And I am feeling very happy to writing about this, so let's starts-

We are going to explain this phenomenon
Explaination is as follows:

First of all do you see any sign that you can ensure that earth is rotating excluding day and night... here a experiment is given to check the rotation of the earth.

Take four balloons and tie it to the surface of the earth by a thread,and fill the normal atmospheric gases in three balloons and…

The ultimate arithmetic formulae on PERCENTAGE for all competitive examination//

In "laws of nature" today we are going to talk about the some important arithmetic formulae on PERCENTAGE, which may be very helpful to you , in all types of competitive examination.

Some important sutras

* If two values are respectively x% and y% more than the third value,then the first is the = (100+x/100+y)×100%
of the second.

* If the two values are respectively x% and y% more than the third value,then second is the=(100+y/100+x)×100% of the first.

* If the two values are respectively x% and y% less than the third value,then the second is the= (100-y/100-x)×100%of the first.

* If the two values are respectively x% and y% less than the third value,then the first is the=(100-x/100-y)×100% of the second.

* If A is x% of C and B is y% of C then A is =x/y×100% of B

* If x% of quantity is taken by the first,y% of remaining is taken by the second and z% of remaining is taken by the third person,now if ₹A is left in the fund, then the fund was in the beginning=
A×100×100×100/(10…

WHY DO PLANETS REVOLVE AROUND THE STARS IN ELLIPTICAL ORBIT.

आज हम लॉज ऑफ नेचर में एक ऐसे क्वेशचन की बात करने जा रहे है,जिसका उत्तर शायद ही किसी को सटीक तरीके से पता है,और ये एस्ट्रोफिजिक्स का एक बहुत ही मजेदार सवाल है कि आखिर कोई प्लैनेट किसी स्टार के चारो तरफ एलिप्टिकल ऑर्बिट में क्यूं रिवोल्व करता है।
लेकिन जब कोई गणितीय डेरिवेशन की बात आती है तब हम अक्सर प्लैनेट के ऑर्बिट को एक परफेक्ट सर्कल मानते है, ग्रेविटेशन चैप्टर में सेटेलाइट की ऑर्बिटल वेलोसिटी का डेरिवेशन करने के लिए हम प्लैनेट और सेटेलाइट के ग्रैविटेशनल फोर्स को उस प्लैनेट के सेट्रीपेटल फोर्स के इक्वल में रखते हैं और वेलोसिटी को निकाल लेते है,वो भी आसानी से, शायद हम ऐसा इसलिए करते है क्यूंकी हमारा कैलकुलेशन करना आसान हो जाता है।
क्या सर्कुलर ऑर्बिट और एलिपटिकल ऑर्बिट में कोई अंतर नहीं है,यदि है तो कहीं न कहीं हमारा कैलकुलेशन जरूर गलत है,क्यूंकि एक एलिप्स और सर्कल के एक्वेशन में काफी अंतर होता है और अलग भी होता है,आप इस अंतर को नीचे साफ देख सकते हैं।
खैर ये छोड़िए आइए हम बात करते है कि आखिर एक प्लैनेट किसी स्टार के चारो तरफ एलिप्टिकल ऑर्बिट में क्यूं घूमता है।
वो कंप्लीट सर्कुलर पा…

GOOGOLPLEX A number which is beyond our thinking.

कहते हैं हमारा ब्रह्मांड अनंत है,और इसकी रचना बिग बैंग के एक विस्फोट से हुआ है,किन्तु सत्य तो ये है कि इस ब्रह्मांड का ना तो सृजन किया जा सकता है और ना ही इसका विनाश किया जा सकता है,
यदि मैं आपसे से पूछूं की ये हमारे अनंत ब्रह्मांड में कुल कितने एटम्स हैं,तब अब क्या कहेंगे,शायद आप के पास कोई शब्द नहीं होगा कुछ कहने के लिए,और होना भी नहीं चाहिए क्यूंकि इस अनंत ब्रह्मांड में कितने एटम्स होंगे इस बात की आप कभी कल्पना भी नहीं कर सकते।
यदि मैं आपसे कहूं की एक ऐसी संख्या भी है जिसको यदि इस ब्रह्मांड के हर एटम्स पर लिखना शुरू किया गया तो शायद इस ब्रह्मांड में एटम्स कम पड़ जाए इसको पूरा लिखने के लिए।
आप शायद हैरान हो गए होंगे और सोच रहे होंगे कि क्या सच ऐसा में है,यदि आप सोच रहे हैं तो बिल्कुल ऐसा ही है।
आज हम लॉज आफ नेचर में एक ऐसी संख्या के बारे में जानकारी प्राप्त करने वाले हैं जो इतना बड़ा हैं कि आप इसकी कल्पना भी नहीं कर सकते हैं।
हम जिस नंबर की बात कर रहे हैं वो नंबर है, 10^(10^100) जिसे गूगोलप्लेक्स कहते है,जिसमें 10^100 को गूगोल कहा जाता है,इसका मतलब है कि एक गूगोलपलेक्स होता है, 10^…