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

Animal and Plant Cells

 Cells Cells are the basic functional, biological and structural unit of life. The word cell is a Latin word meaning ‘small room’. Cells are also known as building blocks of life.  The branch of science that deals with the form, structure, and composition of a cell is known as Cytology. All organisms around us are made up of cells. Bacteria, ameba, paramecium, algae, fungi, plants and animals are made up of cells.  Cells together form tissues. And tissue together makes an organ. History Of Cell The cell was discovered by Robert Hooke in 1665. He assembled a simple microscope and observed a very thin slice of cork under his primitive microscope. The cork was obtained from the outer covering of a tree called bark. Robert Hooke observed many little-partitioned boxes or compartments in the cork slice. These boxes appeared like a honey-comb. He termed these boxes as the cell. He also noticed that one box was separated from another by a wa...
23 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

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

The Inverse & Reciprocal TRIGONOMETRIC Functions

So, this is my second post on trigonometry. In this post we're gonna cover the reciprocal and the inverse Trigonometric functions. If you haven't seen my first post you should definitely view it as it covers the basics of Trigonometry The Reciprocal Trigonometric Functions The reciprocal Trigonometric function of Sine is Cosecant, of Cosine is Secant & for Tangent it is Cotangent. Cosecant (Csc θ = 1/Sin θ) or (Hypotenuse/Opposite) Secant (Sec θ = 1/Cos θ) or (Hypotenuse/Adjacent) Cotangent (Cot θ = 1/Tan θ) or (Adjacent/Opposite) We can also represent Tan θ in another way. As Tan θ = opposite/adjacent  & Sin θ = opposite/hypotenuse  & Cos θ = adjacent/hypotenuse ∴ Tan θ = Sin θ/Cos θ (The hypotenuses cancel out) As Cot θ = 1/Tan θ  So, we can also represent Cot θ as Cos θ/Sin θ.
2 comments
Read more

Catalysts

Catalysts When we hear the word catalyst, the first thing that comes to mind is the game “Mirror’s Edge Catalyst”. But actually, catalysts are chemical substances that speed up the process of a chemical reaction but do not used up in the process of speeding up. The process of using catalysts to speed up chemical processes is called catalysis. Some examples are: Ø Hydrogen peroxide decomposes to form water and oxygen. 2H2O2 à 2H2O + O2 But in the presence of manganese dioxide (MnO2) the process is sped up and happens a lot faster. Ø Cars use a catalytic converter to convert carbon monoxide to carbon dioxide. These contain Platinum to speed up the process and keep the car’s system getting backed up. Ø Ammonia synthesis also uses Iron as a catalyst Catalysts work by reacting with a reactant. This is called a catalytic action. The product of the catalytic action is a chemical intermediate, which can react with the other reactant at a faster pace and give...
Post a Comment
Read more

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