Skip to main content

2-D & 3-D GEOMETRY

2-D & 3-D GEOMETRY

We all have some amount of geometry. We know that any line can be represented on the Cartesian plane. Any figure can be drawn on it. But can we represent a 3-d object on it. Yes we can. A Cartesian plane has 2 axis. While representing in 3-D we need to add a third axis. This axis does not come in between the axis or in the same plane. It appears to be coming out of the paper as we cannot represent a 3-d object on a 2-d surface.


This new z-axis represents a line coming out of the screen. Before understanding 3-d geometry you need to imagine this axis coming out of the screen. 

REMEMBER: all the three axis are perpendicular .i.e there an angle 0f 90 between them and they meet at the origin
If you are unable to imagine you can take a thick book as an example. Any corner becomes it origin and the three edges as the three axis

REPRESENTING 3-D GEOMETRY

Like in 2-d geometry we represent the value of the different axis as (x,y) we use the same method to write in 3-d geometry. (x,y,z). You can imagine that more the value of z the more it will come out of the paper.

SIGN CONVENTIONS IN GEOMETRY

In 2-d geometry we have 4 quadrants or the x-axis and the y-axis divide the plane into 4 parts. In each part different quadrants have different conventions. They are in the following way



In 3-d geometry we have octants that is we have 8 parts. Dividing a cube from the  center to form 8 parts.


We take xy-plane as the surface and z-axis as the height. Remember you need to imagine the above diagram as 3-d
Sign conventions are as following 
1 octant – (X,Y,Z) 
2 octant – (-X,Y,Z)
3 octant – (-X,-Y,Z)
4 octant – (X,-Y,Z)
5 octant – (X,Y,-Z)
6 octant – (-X,Y,-Z)
7 octant – (-X,-Y,-Z)
8 octant – (X,-Y,-Z)

LOCATING A POINT

While locating a point in xy-plane we measure the distances from the axis. The distance away from the y-axis is our x-coordinate and distance from x-axis becomes our y-coordinate. 
   


In 3-d geometry the same method follows but with height. In the below diagram it seems that the distance from x axis is not perpendicular but it is. As I told earlier we cannot exactly show 3-d we can only give 3-d effects 


DISTANCE BETWEEN TWO POINTS

  Let there be two points A and B. they are plotted in a xy-plane then how do we calculate their distance. We use Pythagoras theorem to find the formula. 

Now ,
OL= x1, OM = x2, AL = y1, BM = y2
Therefore 
AC=LM=OM-OL= x2 - x1
BC=BM-BC=BM-AL=y2-y1
Since ABC is a right-triangle then 
AB2 = AC2 + BC2
AB = equation.pdf
Similarly the formula of distance in 3-d is
 equation_1.pdf

SECTION FORMULA 

Let there be line AB. A point P divides the line in the ratio m: n. If coordinates of the two points (A and B) then what will be the coordinates of the P. To find this we use the below section formula . 


equation_2.pdf=equation_3.pdf=equation_4.pdf 
AK=LM=OM – OL= x – x1
PT=MN=ON – OM= x2 – x 
PK=MP – AL= y – y1
BT= BN – PM= y2 – y
equation_5.pdf=equation_6.pdf=equation_7.pdf=equation_8.pdf
Therefore
equation_9.pdf=equation_10.pdf
mx2 – mx = nx – nx1
mx + nx = mx2 + nx1
x(m + n) = mx2 + nx1
x = equation_11.pdf
Similarly
y = equation_12.pdf 
in a 3-d graph the formula remains the same but the ‘x’ becomes z ..that is
z = equation_13.pdf 

CENTROID OF A TRIANGLE

Let there be a triangle with vertices A, B, and C. let AD be the median of the triangle and let the centroid be G. We know that centroid divides the median in the ratio 2: 1. Also D divides the line BC in the ratio 1: 1.

D=equation_14.pdf,equation_15.pdf
D=equation_16.pdf,equation_17.pdf
G =equation_18.pdf,equation_19.pdf
G =equation_20.pdf,equation_21.pdf
Similarly, in 3-d the third axis z comes in the formula
G =equation_22.pdf,equation_23.pdf,equation_24.pdf

AREA OF TRIANGLE

Let there be a triangle ABC. Coordinates of all the vertices are given. So how we can find its area. We can do so by using the formula of trapezium .i.e
Area of trapezium=1/2(sum of parallel sides) X (distance between them)
Now area of ABC will be:-
Area (ABC) = Area (ABLM) + Area (ACNM) – Area (BCNL)
Area (ABC) = ½ [(BL + AM) (LM) + (AM + CN)(MN) – (BL + CN)(LN)
Area (ABC) = ½ [(y2 + y1)(x1 - x2) + (y1 + y3)(x3 – x1) - (y2 + y3)(x3 - x2) ]
Area (ABC) =½ [y2 x1 - y2 x2 + y1 x1 – y1 x2 + y1 x3 – y1 x1 + y3 x3 – y3 x1 - y2 x3 + y2 x2 - y3 x3 + y3 x2]

Labels:

Comments

  1. Sophia mylles2 July 2019 at 17:41

    It’s a good shame you don’t contain a give money button! I’d definitely give money for this fantastic webpage! That i suppose for the time being i’ll be satisfied bookmarking together with including an individual’s Feed that will my best Msn balance. That i appearance forward that will recent messages and definitely will share the web site utilizing my best Facebook or twitter team: ) Pakistan English news

    ReplyDelete
  2. Craton Software Inc.17 October 2019 at 15:00

    I admire this article for the well-researched content and excellent wording. I got so involved in this material that I couldn’t stop reading. I am impressed with your work and skill. Thank you so much. If anyone looking for the math homework help online, Coursecap is the best choice.

    ReplyDelete
  3. Richa Sharma9 April 2020 at 17:06

    Hi there! Someone in my Myspace group shared this site with us so I came to give it a look. I’m definitely loving the information. I’m bookmarking and will be tweeting this to my followers! Outstanding blog and wonderful style and design.
    Sindh News

    ReplyDelete

Post a Comment

Popular Posts

Levitation 2

LEVITATION II To be completely honest I was going to start this with a pun. I did think of one but it doesn’t float… I am sorry I just had to. Anyway, this is the second part to the article on super cool ways of making things levitate. Go check the first part out if you haven’t already. Actually, the first part may have become repulsive with all the magnets and stuff, but I promise this will be more attractive. Get it? No? I’ll stop now. I am just going to jump straight into it. 1.    Electrostatic Levitation I know you are probably sick and tired of magnets but they are the best way you know… This method is somewhat similar. You remember that cool science experiment you did with two straws attracting or repulsing each other based on their charge? So basically using the same principle we can make a charged object levitate. But before you try it, let me tell you it won’t be easy. Even impossible according to our Mr. Earnshaw. He even made a law (th...
4 comments
Read more

Paid Apps For Free ??

Everyone wants to play a game like GTA on iPad, it is easy on a computer to download such games but difficult on smart devices like iPad, tablet,  or smartphones. We can buy them but not everyone can buy games. But no worries guys there is a solution to this problem where one can have fun of playing games without spending their precious money .(underlined apps have downloading links  given at the end) iOS There many apps through which you can download these paid games for free. Also, there are many sites for the same. One of the most helpful apps is a Chinese app.It is called haimawan. If on a ios device, then you just have to click install which will redirect to settings  add a profile  and boom enjoy it as you wish . but it does not always work, it might work for few days and then the verification problem which may not cure. One more app is tutu which is a fantastic app and always work (99.9% sure .. nothing is perfect). X cross was the perfect app b...
14 comments
Read more

Rutherford Alpha Ray Scattering Experiment

Rutherford Alpha Ray Scattering Experiment Hey, Guys, most of you might have heard about the alpha ray scattering experiment and if you want to know in detail about Rutherford's model and the experiment he conducted, this is the right place for you... But first: Things You Must Know Some basic information that will help you understand rutherford experiment properly: Proton is a sub-atomic particle which is positively charged and has a mass of 1u. Alpha particles are helium atom with a charge of +2 as they have lost 2 electrons. Alpha particles have an atomic mass  of 4u. Gold is highly malleable and can be beaten into very thin sheets. Experiment Rutherford conducted his experiment in the following way: Rutherford took a very thin gold foil and bombarded it with high energy alpha particles. He placed a layer of zinc sulphide on the walls where the experiment was taking place because when alpha particles ...
10 comments
Read more

Permutation and Combination

Permutation and Combination Hey guys, I am back with one more interesting mathematics topic. And I am pretty sure that you guys would find it interesting too because many of you must have faced problems regarding these arrangements. Although if you were not able to solve them at that moment of time I am sure after completing this article you would solve the problem in minutes or even seconds. WHAT IS THE DIFFERENCE ?? Many people get confused between these two terms permutation and combination. They both have almost similar use but have a vast difference in their meaning. You may think that both of them mean arranging entities, then what is the difference ?? For making it easy we will take an example, lets say I have four friends : Arya, Bhavesh, Chirayu and Dhruv. So if I arrange them like Arya, Bhavesh, Chirayu and Dhruv or Dhruv, Chirayu, Bhavesh and Arya, it will make no difference in combinations but if you check them according to permutation they will be different. As ...
5 comments
Read more

Important Mathematical Constants!

Important Mathematical Constants Mathematical constants are those numbers that are special and interesting because they come up in the various fields of mathematics like geometry, calculus etc. These mathematical constants are usually named after the person who discovered it and they are represented by a symbol that is usually picked up from the Greek alphabet. Mathematical constants are by definition very important. In this article we will take a look at certain mathematical constants that are more commonplace than others. 1.       π (pi) or Archimedes constant (~3.14159):   π is defined as the ratio of the circumference of a circle to its diameter. This is probably the most popular mathematical constant. So π is the circumference of the circle whose diameter is 1 unit. You might have seen it popping up when calculating the area of a circle (πr 2 ) or the circumference of a circle (2πr). It has many uses throughout mathematics from calcula...
4 comments
Read more

High School Pedia

It is an initiative by some students to spread the light of knowledge to everyone and everywhere. It was started in the year 2015 and has grown rapidly in the past few months. By the means of this website, we try to provide information on every topic that we can reach up to. You can find different articles on this website. All these articles are written in simple language so that everyone can understand it and learn from it. We at High School Pedia believe in creative learning and this is the reason why we add our own edited graphical representations in every article. Once a very learned man said, “Knowledge increases by not keeping it to yourself but by sharing it with others”. And we follow the same motto “Share to Learn”. The team of High School Pedia tries its best to provide you with the best and original content. Unlike many other websites, High School Pedia is famous for its original and inspiring content.
2 comments
Read more

Finding Square Roots Easily (Vedic Math)

Finding Square Roots Easily (Vedic Math) Hello, Guys!!! I’m back with another post on Vedic Math that is going to keep you glued rooted to your spot 😆. This time I’m going to teach you how to calculate the square roots of perfect squares faster… With a little practice, this method can let you calculate the roots in a matter of 5 seconds, whereas if you had used the traditional method, you would still be stuck with your question 😏 Things you need to know:- Perfect square which has unit digit 1 will have square root with the last digit as 1 or 9 Perfect square which has unit digit 4 will have square root with the last digit as 2 or 8 Perfect square which has unit digit 5 will have square root with the last digit as 5 Perfect square which has unit digit 6 will have square root with the last digit as 4 or 6 Perfect square which has unit digit 9 will have square root with the last digit as 3 or 7 Perfect square which has unit digit 0 will have square r...
Post a Comment
Read more