# 2.2 Vectors in three dimensions

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• Describe three-dimensional space mathematically.
• Locate points in space using coordinates.
• Write the distance formula in three dimensions.
• Write the equations for simple planes and spheres.
• Perform vector operations in ${ℝ}^{3}.$

Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. Does your planned route go through the mountains? Do you have to cross a river? To appreciate fully the impact of these geographic features, you must use three dimensions. This section presents a natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.

## Three-dimensional coordinate systems

As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal x -axis and the vertical y -axis. We can add a third dimension, the z -axis, which is perpendicular to both the x -axis and the y -axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.

## Definition

The three-dimensional rectangular coordinate system    consists of three perpendicular axes: the x -axis, the y -axis, and the z -axis. Because each axis is a number line representing all real numbers in $ℝ,$ the three-dimensional system is often denoted by ${ℝ}^{3}.$

In [link] (a), the positive z -axis is shown above the plane containing the x - and y -axes. The positive x -axis appears to the left and the positive y -axis is to the right. A natural question to ask is: How was arrangement determined? The system displayed follows the right-hand rule    . If we take our right hand and align the fingers with the positive x -axis, then curl the fingers so they point in the direction of the positive y -axis, our thumb points in the direction of the positive z -axis. In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation.

In two dimensions, we describe a point in the plane with the coordinates $\left(x,y\right).$ Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, $z,$ is appended to indicate alignment with the z -axis: $\left(x,y,z\right).$ A point in space is identified by all three coordinates ( [link] ). To plot the point $\left(x,y,z\right),$ go x units along the x -axis, then $y$ units in the direction of the y -axis, then $z$ units in the direction of the z -axis.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?