2.5 Equations of lines and planes in space  (Page 2/19)

 Page 2 / 19

Parametric and symmetric equations of a line

A line $L$ parallel to vector $\text{v}=⟨a,b,c⟩$ and passing through point $P\left({x}_{0},{y}_{0},{z}_{0}\right)$ can be described by the following parametric equations:

$x={x}_{0}+ta,y={y}_{0}+tb,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}z={z}_{0}+tc.$

If the constants $a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c$ are all nonzero, then $L$ can be described by the symmetric equation of the line:

$\frac{x-{x}_{0}}{a}=\frac{y-{y}_{0}}{b}=\frac{z-{z}_{0}}{c}.$

The parametric equations of a line are not unique. Using a different parallel vector or a different point on the line leads to a different, equivalent representation. Each set of parametric equations leads to a related set of symmetric equations, so it follows that a symmetric equation of a line is not unique either.

Equations of a line in space

Find parametric and symmetric equations of the line passing through points $\left(1,4,-2\right)$ and $\left(-3,5,0\right).$

First, identify a vector parallel to the line:

$\text{v}=⟨-3-1,5-4,0-\left(-2\right)⟩=⟨-4,1,2⟩.$

Use either of the given points on the line to complete the parametric equations:

$x=1-4t,y=4+t,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}z=-2+2t.$

Solve each equation for $t$ to create the symmetric equation of the line:

$\frac{x-1}{-4}=y-4=\frac{z+2}{2}.$

Find parametric and symmetric equations of the line passing through points $\left(1,-3,2\right)$ and $\left(5,-2,8\right).$

Possible set of parametric equations: $x=1+4t,y=-3+t,z=2+6t;$

related set of symmetric equations: $\frac{x-1}{4}=y+3=\frac{z-2}{6}$

Sometimes we don’t want the equation of a whole line, just a line segment. In this case, we limit the values of our parameter $t.$ For example, let $P\left({x}_{0},{y}_{0},{z}_{0}\right)$ and $Q\left({x}_{1},{y}_{1},{z}_{1}\right)$ be points on a line, and let $\text{p}=⟨{x}_{0},{y}_{0},{z}_{0}⟩$ and $\text{q}=⟨{x}_{1},{y}_{1},{z}_{1}⟩$ be the associated position vectors. In addition, let $\text{r}=⟨x,y,z⟩.$ We want to find a vector equation for the line segment between $P$ and $Q.$ Using $P$ as our known point on the line, and $\stackrel{\to }{PQ}=⟨{x}_{1}-{x}_{0},{y}_{1}-{y}_{0},{z}_{1}-{z}_{0}⟩$ as the direction vector equation, [link] gives

$\text{r}=\text{p}+t\left(\stackrel{\to }{PQ}\right).$

Using properties of vectors, then

$\begin{array}{cc}\hfill \text{r}& =\text{p}+t\left(\stackrel{\to }{PQ}\right)\hfill \\ & =⟨{x}_{0},{y}_{0},{z}_{0}⟩+t⟨{x}_{1}-{x}_{0},{y}_{1}-{y}_{0},{z}_{1}-{z}_{0}⟩\hfill \\ & =⟨{x}_{0},{y}_{0},{z}_{0}⟩+t\left(⟨{x}_{1},{y}_{1},{z}_{1}⟩-⟨{x}_{0},{y}_{0},{z}_{0}⟩\right)\hfill \\ & =⟨{x}_{0},{y}_{0},{z}_{0}⟩+t⟨{x}_{1},{y}_{1},{z}_{1}⟩-t⟨{x}_{0},{y}_{0},{z}_{0}⟩\hfill \\ & =\left(1-t\right)⟨{x}_{0},{y}_{0},{z}_{0}⟩+t⟨{x}_{1},{y}_{1},{z}_{1}⟩\hfill \\ & =\left(1-t\right)\text{p}+t\text{q}.\hfill \end{array}$

Thus, the vector equation of the line passing through $P$ and $Q$ is

$\text{r}=\left(1-t\right)\text{p}+t\text{q}.$

Remember that we didn’t want the equation of the whole line, just the line segment between $P$ and $Q.$ Notice that when $t=0,$ we have $r=p,$ and when $t=1,$ we have $r=q.$ Therefore, the vector equation of the line segment between $P$ and $Q$ is

$\text{r}=\left(1-t\right)\text{p}+t\text{q},0\le t\le 1.$

Going back to [link] , we can also find parametric equations for this line segment. We have

$\begin{array}{ccc}\hfill \text{r}& =\hfill & \text{p}+t\left(\stackrel{\to }{PQ}\right)\hfill \\ \hfill ⟨x,y,z⟩& =\hfill & ⟨{x}_{0},{y}_{0},{z}_{0}⟩+t⟨{x}_{1}-{x}_{0},{y}_{1}-{y}_{0},{z}_{1}-{z}_{0}⟩\hfill \\ & =\hfill & ⟨{x}_{0}+t\left({x}_{1}-{x}_{0}\right),{y}_{0}+t\left({y}_{1}-{y}_{0}\right),{z}_{0}+t\left({z}_{1}-{z}_{0}\right)⟩.\hfill \end{array}$

Then, the parametric equations are

$x={x}_{0}+t\left({x}_{1}-{x}_{0}\right),y={y}_{0}+t\left({y}_{1}-{y}_{0}\right),z={z}_{0}+t\left({z}_{1}-{z}_{0}\right),0\le t\le 1.$

Parametric equations of a line segment

Find parametric equations of the line segment between the points $P\left(2,1,4\right)$ and $Q\left(3,-1,3\right).$

$x={x}_{0}+t\left({x}_{1}-{x}_{0}\right),y={y}_{0}+t\left({y}_{1}-{y}_{0}\right),z={z}_{0}+t\left({z}_{1}-{z}_{0}\right),0\le t\le 1.$

Working with each component separately, we get

$\begin{array}{cc}\hfill x& ={x}_{0}+t\left({x}_{1}-{x}_{0}\right)\hfill \\ & =2+t\left(3-2\right)\hfill \\ & =2+t,\hfill \end{array}$
$\begin{array}{cc}\hfill y& ={y}_{0}+t\left({y}_{1}-{y}_{0}\right)\hfill \\ & =1+t\left(-1-1\right)\hfill \\ & =1-2t,\hfill \end{array}$

and

$\begin{array}{cc}\hfill z& ={z}_{0}+t\left({z}_{1}-{z}_{0}\right)\hfill \\ & =4+t\left(3-4\right)\hfill \\ & =4-t.\hfill \end{array}$

Therefore, the parametric equations for the line segment are

$x=2+t,y=1-2t,z=4-t,0\le t\le 1.$

Find parametric equations of the line segment between points $P\left(-1,3,6\right)$ and $Q\left(-8,2,4\right).$

$x=-1-7t,y=3-t,z=6-2t,0\le t\le 1$

Distance between a point and a line

We already know how to calculate the distance between two points in space. We now expand this definition to describe the distance between a point and a line in space. Several real-world contexts exist when it is important to be able to calculate these distances. When building a home, for example, builders must consider “setback” requirements, when structures or fixtures have to be a certain distance from the property line. Air travel offers another example. Airlines are concerned about the distances between populated areas and proposed flight paths.

Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
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
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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!