# 2.2 Vectors in three dimensions

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• Describe three-dimensional space mathematically.
• Locate points in space using coordinates.
• Write the distance formula in three dimensions.
• Write the equations for simple planes and spheres.
• Perform vector operations in ${ℝ}^{3}.$

Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. Does your planned route go through the mountains? Do you have to cross a river? To appreciate fully the impact of these geographic features, you must use three dimensions. This section presents a natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.

## Three-dimensional coordinate systems

As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal x -axis and the vertical y -axis. We can add a third dimension, the z -axis, which is perpendicular to both the x -axis and the y -axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.

## Definition

The three-dimensional rectangular coordinate system    consists of three perpendicular axes: the x -axis, the y -axis, and the z -axis. Because each axis is a number line representing all real numbers in $ℝ,$ the three-dimensional system is often denoted by ${ℝ}^{3}.$

In [link] (a), the positive z -axis is shown above the plane containing the x - and y -axes. The positive x -axis appears to the left and the positive y -axis is to the right. A natural question to ask is: How was arrangement determined? The system displayed follows the right-hand rule    . If we take our right hand and align the fingers with the positive x -axis, then curl the fingers so they point in the direction of the positive y -axis, our thumb points in the direction of the positive z -axis. In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation. (a) We can extend the two-dimensional rectangular coordinate system by adding a third axis, the z -axis, that is perpendicular to both the x -axis and the y -axis. (b) The right-hand rule is used to determine the placement of the coordinate axes in the standard Cartesian plane.

In two dimensions, we describe a point in the plane with the coordinates $\left(x,y\right).$ Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, $z,$ is appended to indicate alignment with the z -axis: $\left(x,y,z\right).$ A point in space is identified by all three coordinates ( [link] ). To plot the point $\left(x,y,z\right),$ go x units along the x -axis, then $y$ units in the direction of the y -axis, then $z$ units in the direction of the z -axis. To plot the point ( x , y , z ) go x units along the x -axis, then y units in the direction of the y -axis, then z units in the direction of the z -axis.

#### Questions & Answers

how can chip be made from sand
Eke Reply
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Almas
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Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
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currently
William
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Maira Reply
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learn
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learn
Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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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.
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
Brian Reply
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
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LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
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Mahi
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Rafiq
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Anam
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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Bob Reply
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
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

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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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