<< Chapter < Page Chapter >> Page >
  • 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, u and v , are parallel, we claim there must be a scalar, k , such that u = k v . If u and v have the same direction, simply choose k = u v . If u and v have opposite directions, choose k = u v . Note that the converse holds as well. If u = k v for some scalar k , then either u and v have the same direction ( k > 0 ) or opposite directions ( k < 0 ) , so u and v are parallel. Therefore, two nonzero vectors u and v are parallel if and only if u = k v for some scalar 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 ( x 0 , y 0 , z 0 ) . Let v = a , b , c be a vector parallel to L . Then, for any point on line Q ( x , y , z ) , we know that P Q is parallel to v . Thus, as we just discussed, there is a scalar, t , such that P Q = t v , which gives

P Q = t v x x 0 , y y 0 , z z 0 = t a , b , c x x 0 , y y 0 , z z 0 = t a , t b , t c .
This figure is the first octant of the 3-dimensional coordinate system. There is a line segment passing through two points. The points are labeled “P = (x sub 0, y sub 0, z sub 0)” and “Q = (x, y, z).” There is also a vector in standard position drawn. The vector is labeled “v = <a, b, c>.”
Vector v is the direction vector for P Q .

Using vector operations, we can rewrite [link] as

x x 0 , y y 0 , z z 0 = t a , t b , t c x , y , z x 0 , y 0 , z 0 = t a , b , c x , y , z = x 0 , y 0 , z 0 + t a , b , c .

Setting r = x , y , z and r 0 = x 0 , y 0 , z 0 , we now have the vector equation of a line    :

r = r 0 + t v .

Equating components, [link] shows that the following equations are simultaneously true: x x 0 = t a , y y 0 = t b , and z z 0 = t c . If we solve each of these equations for the component variables x , y , and 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 + t a y = y 0 + t b z = z 0 + t c .

If we solve each of the equations for t assuming a , b , and c are nonzero, we get a different description of the same line:

x x 0 a = t y y 0 b = t 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    :

x x 0 a = y y 0 b = z z 0 c .

We summarize the results in the following theorem.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 9

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?