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English Guide: Practical Tips for Better English Communication

article thumbnailGood teamwork is essential not only for the growth of any organisation but also for the personal growth of individuals in the organisation. An essential component of achieving good team bonding is when team members express positive thoughts about fellow team members to them directly.  Sincere compliments and praise go a long way in making a person feel good and feel motivated to perform better for the team and the organisation.     There are a...
article thumbnailAt the outset, let us state that we hope you never have to apply for a compassionate leave. However, the reality is that most of us, at some stage of our life, end up having to request for such leaves. This article is written to specifically help you in case of such an eventuality.     Before we get into the mechanics of applying for compassionate leave, let us first understand what is compassionate leave. It is leave granted to an employee upon...
article thumbnailThe use of idioms in regular communication serves to add more ‘colour’ to the communication and makes it more interesting and engaging. Therefore, it is quite common to see the usage of lots of English idioms, especially sports-related idioms, in any office communication. As a young professional, adapting yourself to an office environment, it will help to not only be familiar with some of the common idioms but also learn to use them in your...
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article thumbnailThis is another article written specifically to help visitors to our blog avoid some common grammar mistakes and improve their spoken English skills. In this piece, we will focus on a common mistake that we keep hearing and reading from so many English speakers these days: the use of “one of the….” in sentences.   To start with, let us take some examples of the type of sentences that you may come across in everyday conversation. Pay specific...
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Top grammar mistakes to avoid in spoken English Do you know that English is a confusing language? It is. Well, very confusing, but there are many rules that guide us as to how and when to use some words and its derivatives  that can make it less confusing. While learning words in English, building your vocabulary and mastering correct pronunciation is not a major task, learning how to use the words correctly with a proper grammar sense is what separates an amateur English speaker from a...
Social etiquette: How to politely interrupt a conversation Isn’t it annoying when somebody interrupts you while you are speaking to someone, especially if you are discussing something rather important or are in the middle of a conversation or an argument. But sometimes interruption becomes mandatory and people have to do it; also there will be instances where you will need to interrupt a person, and you definitely don’t want to be annoying. So how do you interrupt a conversation politely? How do you...
Popular Brand Names You’ve Been Mispronouncing All This Time Imagine this – you are amidst a number of people, just talking and chit-chatting, and suddenly you guys start talking about things you recently shopped, and you are excited to tell everyone about the cute, beautiful Chanel lipstick that you have bought. After all, you have blown up a fortune to buy those, it is only natural to talk about it.So, you say, ‘I love the new Chanel lipstick that I bought, it has the most amazing colour, and totally...

Angle between two planes

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Angle between two planes - Introduction    

ang_plane_1
The angle between two planes is defined as the angle between their normals If θ is the angle between two planes, then so is 180 - θ. We shall take the acute angle as the angle between two planes



Vector Form:
  If r.n1=d1 and r.n2=d2 are the equation of two planes then angle between them is given by the equation

                       ang_plane_3

                             ang_plane_4

Cartesian Form:
  If A1x + B1y + C1z + D1=0 and A2x + B2y + C2z + D2=0 are the Cartesian equations of two planes and θ is the angle between them then


                    ang_plane_5



Condition for parallelism and perpendicularity

1. If the planes are parallel then ang_plane_6
2. If the planes are perpendicular then A1A2 + B1B2 + C1C2=0



Coplanarity of Two Lines

Vector Form:  If r = a1 + λb1 and r= a2 + μb2 are the equations of two lines then they are said to be coplanar if (a2-a1).(b1 x b2)=0

Cartesian Form:  If A(x1, y1, z1) and B(x2, y2, z2) are two points with the direction ratios of parallel vectors <a1, b1, c1> and <a2, b2, c2>, then the lines are said to be coplanar if
         
               ang_plane_7



Distance of a point from a plane

Vector Form:  If the equation of the plane is in the form r.N=d, where N is normal to the plane, then the perpendicular distance is ang_plane_8

The length of perpendicular from origin O to the plane r.N=d is |d|/|N|

Cartesian Form:
  If P(x1, y1, z1) be the given point with position vector a and Ax + By + Cz=D be the equation of the plane then the perpendicular distance from P to the plane is given by d= ang_plane_13



Angle between a Line and a Plane

If r=ab be the equation of the line and r.n=d be the equation of the plane the angle between them is given by
                              ang_plane_10

                               ang_plane_11


Example:
Find the distance of a point (2, 5, -3) from the plane 6x - 3y + 2z - 4=0


Solution:
Distance ang_plane_12


                            ang_plane_2
                      
                            = 13/7



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