2.2 Vectors in three dimensions  (Page 2/14)

 Page 2 / 14

Locating points in space

Sketch the point $\left(1,-2,3\right)$ in three-dimensional space.

To sketch a point, start by sketching three sides of a rectangular prism along the coordinate axes: one unit in the positive $x$ direction, $2$ units in the negative $y$ direction, and $3$ units in the positive $z$ direction. Complete the prism to plot the point ( [link] ). Sketching the point ( 1 , −2 , 3 ) .

Sketch the point $\left(-2,3,-1\right)$ in three-dimensional space. In two-dimensional space, the coordinate plane is defined by a pair of perpendicular axes. These axes allow us to name any location within the plane. In three dimensions, we define coordinate planes by the coordinate axes, just as in two dimensions. There are three axes now, so there are three intersecting pairs of axes. Each pair of axes forms a coordinate plane: the xy -plane, the xz -plane, and the yz -plane ( [link] ). We define the xy -plane formally as the following set: $\left\{\left(x,y,0\right):x,y\in ℝ\right\}.$ Similarly, the xz -plane and the yz -plane are defined as $\left\{\left(x,0,z\right):x,z\in ℝ\right\}$ and $\left\{\left(0,y,z\right):y,z\in ℝ\right\},$ respectively.

To visualize this, imagine you’re building a house and are standing in a room with only two of the four walls finished. (Assume the two finished walls are adjacent to each other.) If you stand with your back to the corner where the two finished walls meet, facing out into the room, the floor is the xy -plane, the wall to your right is the xz -plane, and the wall to your left is the yz -plane. The plane containing the x - and y -axes is called the xy -plane. The plane containing the x - and z -axes is called the xz -plane, and the y - and z -axes define the yz -plane.

In two dimensions, the coordinate axes partition the plane into four quadrants. Similarly, the coordinate planes divide space between them into eight regions about the origin, called octants    . The octants fill ${ℝ}^{3}$ in the same way that quadrants fill ${ℝ}^{2},$ as shown in [link] .

Most work in three-dimensional space is a comfortable extension of the corresponding concepts in two dimensions. In this section, we use our knowledge of circles to describe spheres, then we expand our understanding of vectors to three dimensions. To accomplish these goals, we begin by adapting the distance formula to three-dimensional space.

If two points lie in the same coordinate plane, then it is straightforward to calculate the distance between them. We that the distance $d$ between two points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ in the xy -coordinate plane is given by the formula

$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}.$

The formula for the distance between two points in space is a natural extension of this formula.

The distance between two points in space

The distance $d$ between points $\left({x}_{1},{y}_{1},{z}_{1}\right)$ and $\left({x}_{2},{y}_{2},{z}_{2}\right)$ is given by the formula

$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}.$

The proof of this theorem is left as an exercise. ( Hint: First find the distance ${d}_{1}$ between the points $\left({x}_{1},{y}_{1},{z}_{1}\right)$ and $\left({x}_{2},{y}_{2},{z}_{1}\right)$ as shown in [link] .) The distance between P 1 and P 2 is the length of the diagonal of the rectangular prism having P 1 and P 2 as opposite corners.

Distance in space

Find the distance between points ${P}_{1}=\left(3,\text{−}1,5\right)$ and ${P}_{2}=\left(2,1,\text{−}1\right).$

Substitute values directly into the distance formula:

$\begin{array}{cc}\hfill d\left({P}_{1},{P}_{2}\right)& =\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}\hfill \\ & =\sqrt{{\left(2-3\right)}^{2}+{\left(1-\left(-1\right)\right)}^{2}+{\left(-1-5\right)}^{2}}\hfill \\ & =\sqrt{{1}^{2}+{2}^{2}+{\left(-6\right)}^{2}}\hfill \\ & =\sqrt{41}.\hfill \end{array}$

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!       By Nick Swain   By