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]

Comments

  1. 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. 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. 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

Anode Ray Experiment

→Anode ray experiment was conducted by E Goldstein. →These rays are also known as canal rays. →This experiment helped in the discovery of the proton. Apparatus Used A discharge tube  was taken in which there were 2 electrodes i.e. Anode(+ve) and the cathode (-ve). The tube was filled with an inert gas. A perforated or porous cathode was used. A layer of zinc sulphide was placed at the back of the cathode. There was a vacuum pump in the tube. High voltage (5000v-10000v) was allowed to flow through the system. It was observed that when the gas was at 1atm(atmospheric pressure ) no change was seen in the tube.  When the   pressure   was decreased inside the tube, a glow could be seen at the back side of the cathode.

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 wall. What Ho

Isotopes, Isobars and Isotones

Isotopes These are elements which have the same atomic number but different atomic mass . They have the same atomic number because the number of protons that are inside their nuclei remains the same. But, they have different atomic mass because the number of neutrons that are also inside their nuclei is different. As the number of protons inside nuclei remains same, therefore the overall charge of the elements also remains same as in isotopes: no of protons = no of electrons . Hence, as isotopes overall charge remains neutral, therefore their chemical properties will also remain identical.   Therefore, Isotopes are chemically same but physically different.

Cathode Ray Experiment

This experiment was conducted by J.J. Thomson (Sir Joseph John Thomson) in the year 1897. This experiment proved that atom is made up of fundamental particles which are much smaller than the smallest atom 'hydrogen' This experiment helped to discover electron. According to J.J. Thomson, the cathode rays consisted of very light, small and negatively charged particles. He named the particles "corpuscles" which were later known as electrons

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 (the law is

Pokemon Go 0.33.0

All Pokemon go fans get ready because again the latest version of Pokemon go is out. The Niantic has released a new version of Pokemon go i.e. 0.33.0.The size of the new version is 61.48MB.

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 strike zinc sulphide layer, it results i