# 2.5 Equations of lines and planes in space

 Page 1 / 19
• Write the vector and parametric equations, and the general form, of a line through a given point in a given direction, and a line through two given points.
• Find the distance from a point to a given line.
• Write the vector and scalar equations of a plane through a given point with a given normal.
• Find the distance from a point to a given plane.
• Find the angle between two planes.

By now, we are familiar with writing equations that describe a line in two dimensions. To write an equation for a line, we must know two points on the line, or we must know the direction of the line and at least one point through which the line passes. In two dimensions, we use the concept of slope to describe the orientation, or direction, of a line. In three dimensions, we describe the direction of a line using a vector parallel to the line. In this section, we examine how to use equations to describe lines and planes in space.

## Equations for a line in space

Let’s first explore what it means for two vectors to be parallel. Recall that parallel vectors must have the same or opposite directions. If two nonzero vectors, $\text{u}$ and $\text{v},$ are parallel, we claim there must be a scalar, $k,$ such that $\text{u}=k\text{v}.$ If $\text{u}$ and $\text{v}$ have the same direction, simply choose $k=\frac{‖\text{u}‖}{‖\text{v}‖}.$ If $\text{u}$ and $\text{v}$ have opposite directions, choose $k=-\frac{‖\text{u}‖}{‖\text{v}‖}.$ Note that the converse holds as well. If $\text{u}=k\text{v}$ for some scalar $\text{k},$ then either $\text{u}$ and $\text{v}$ have the same direction $\left(k>0\right)$ or opposite directions $\left(k<0\right),$ so $\text{u}$ and $\text{v}$ are parallel. Therefore, two nonzero vectors $\text{u}$ and $\text{v}$ are parallel if and only if $\text{u}=k\text{v}$ for some scalar $\text{k}.$ By convention, the zero vector $0$ is considered to be parallel to all vectors.

As in two dimensions, we can describe a line in space using a point on the line and the direction of the line, or a parallel vector, which we call the direction vector    ( [link] ). Let $L$ be a line in space passing through point $P\left({x}_{0},{y}_{0},{z}_{0}\right).$ Let $\text{v}=⟨a,b,c⟩$ be a vector parallel to $L.$ Then, for any point on line $Q\left(x,y,z\right),$ we know that $\stackrel{\to }{PQ}$ is parallel to $\text{v}.$ Thus, as we just discussed, there is a scalar, $t,$ such that $\stackrel{\to }{PQ}=t\text{v},$ which gives

$\begin{array}{ccc}\hfill \stackrel{\to }{PQ}& =\hfill & t\text{v}\hfill \\ \hfill ⟨x-{x}_{0},y-{y}_{0},z-{z}_{0}⟩& =\hfill & t⟨a,b,c⟩\hfill \\ \hfill ⟨x-{x}_{0},y-{y}_{0},z-{z}_{0}⟩& =\hfill & ⟨ta,tb,tc⟩.\hfill \end{array}$

Using vector operations, we can rewrite [link] as

$\begin{array}{ccc}\hfill ⟨x-{x}_{0},y-{y}_{0},z-{z}_{0}⟩& =\hfill & ⟨ta,tb,tc⟩\hfill \\ \hfill ⟨x,y,z⟩-⟨{x}_{0},{y}_{0},{z}_{0}⟩& =\hfill & t⟨a,b,c⟩\hfill \\ \hfill ⟨x,y,z⟩& =\hfill & ⟨{x}_{0},{y}_{0},{z}_{0}⟩+t⟨a,b,c⟩.\hfill \end{array}$

Setting $\text{r}=⟨x,y,z⟩$ and ${\text{r}}_{0}=⟨{x}_{0},{y}_{0},{z}_{0}⟩,$ we now have the vector equation of a line    :

$\text{r}={\text{r}}_{0}+t\text{v}.$

Equating components, [link] shows that the following equations are simultaneously true: $x-{x}_{0}=ta,$ $y-{y}_{0}=tb,$ and $z-{z}_{0}=tc.$ If we solve each of these equations for the component variables $x,y,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}z,$ we get a set of equations in which each variable is defined in terms of the parameter t and that, together, describe the line. This set of three equations forms a set of parametric equations of a line    :

$x={x}_{0}+ta\phantom{\rule{1em}{0ex}}y={y}_{0}+tb\phantom{\rule{1em}{0ex}}z={z}_{0}+tc.$

If we solve each of the equations for $t$ assuming $a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c$ are nonzero, we get a different description of the same line:

$\frac{x-{x}_{0}}{a}=t\phantom{\rule{1em}{0ex}}\frac{y-{y}_{0}}{b}=t\phantom{\rule{1em}{0ex}}\frac{z-{z}_{0}}{c}=t.$

Because each expression equals t , they all have the same value. We can set them equal to each other to create symmetric equations of a line    :

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

We summarize the results in the following theorem.

#### Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

 By By