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  • 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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
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?
Radek Reply
Practice Key Terms 9

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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