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] ).

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.

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] .)

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:

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

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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?