# 2.2 Vectors in three dimensions  (Page 5/14)

 Page 5 / 14

Three-dimensional vectors can also be represented in component form. The notation $\text{v}=⟨x,y,z⟩$ is a natural extension of the two-dimensional case, representing a vector with the initial point at the origin, $\left(0,0,0\right),$ and terminal point $\left(x,y,z\right).$ The zero vector is $0=⟨0,0,0⟩.$ So, for example, the three dimensional vector $\text{v}=⟨2,4,1⟩$ is represented by a directed line segment from point $\left(0,0,0\right)$ to point $\left(2,4,1\right)$ ( [link] ).

Vector addition and scalar multiplication are defined analogously to the two-dimensional case. If $\text{v}=⟨{x}_{1},{y}_{1},{z}_{1}⟩$ and $\text{w}=⟨{x}_{2},{y}_{2},{z}_{2}⟩$ are vectors, and $k$ is a scalar, then

$\text{v}+\text{w}=⟨{x}_{1}+{x}_{2},{y}_{1}+{y}_{2},{z}_{1}+{z}_{2}⟩\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}k\text{v}=⟨k{x}_{1},k{y}_{1},k{z}_{1}⟩.$

If $k=-1,$ then $k\text{v}=\left(-1\right)\text{v}$ is written as $\text{−}\text{v},$ and vector subtraction is defined by $\text{v}-w=v+\left(\text{−}\text{w}\right)=v+\left(-1\right)\text{w}.$

The standard unit vectors extend easily into three dimensions as well— $\text{i}=⟨1,0,0⟩,$ $\text{j}=⟨0,1,0⟩,$ and $\text{k}=⟨0,0,1⟩$ —and we use them in the same way we used the standard unit vectors in two dimensions. Thus, we can represent a vector in ${ℝ}^{3}$ in the following ways:

$\text{v}=⟨x,y,z⟩=x\text{i}+y\text{j}+z\text{k}.$

## Vector representations

Let $\stackrel{\to }{PQ}$ be the vector with initial point $P=\left(3,12,6\right)$ and terminal point $Q=\left(-4,-3,2\right)$ as shown in [link] . Express $\stackrel{\to }{PQ}$ in both component form and using standard unit vectors.

In component form,

$\begin{array}{cc}\hfill \stackrel{\to }{PQ}& =⟨{x}_{2}-{x}_{1},{y}_{2}-{y}_{1},{z}_{2}-{z}_{1}⟩\hfill \\ & =⟨-4-3,-3-12,2-6⟩=⟨-7,-15,-4⟩.\hfill \end{array}$

In standard unit form,

$\stackrel{\to }{PQ}=-7\text{i}-15\text{j}-4\text{k}.$

Let $S=\left(3,8,2\right)$ and $T=\left(2,-1,3\right).$ Express $\stackrel{\to }{ST}$ in component form and in standard unit form.

$\stackrel{\to }{ST}=⟨-1,-9,1⟩=\text{−}\text{i}-9\text{j}+\text{k}$

As described earlier, vectors in three dimensions behave in the same way as vectors in a plane. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space ( [link] ).

We have already seen how some of the algebraic properties of vectors, such as vector addition and scalar multiplication, can be extended to three dimensions. Other properties can be extended in similar fashion. They are summarized here for our reference.

## Rule: properties of vectors in space

Let $\text{v}=⟨{x}_{1},{y}_{1},{z}_{1}⟩$ and $\text{w}=⟨{x}_{2},{y}_{2},{z}_{2}⟩$ be vectors, and let $k$ be a scalar.

Scalar multiplication: $k\text{v}=⟨k{x}_{1},k{y}_{1},k{z}_{1}⟩$

Vector addition: $\text{v}+\text{w}=⟨{x}_{1},{y}_{1},{z}_{1}⟩+⟨{x}_{2},{y}_{2},{z}_{2}⟩=⟨{x}_{1}+{x}_{2},{y}_{1}+{y}_{2},{z}_{1}+{z}_{2}⟩$

Vector subtraction: $\text{v}-\text{w}=⟨{x}_{1},{y}_{1},{z}_{1}⟩-⟨{x}_{2},{y}_{2},{z}_{2}⟩=⟨{x}_{1}-{x}_{2},{y}_{1}-{y}_{2},{z}_{1}-{z}_{2}⟩$

Vector magnitude: $‖\text{v}‖=\sqrt{{x}_{1}{}^{2}+{y}_{1}{}^{2}+{z}_{1}{}^{2}}$

Unit vector in the direction of v: $\frac{1}{‖\text{v}‖}\text{v}=\frac{1}{‖\text{v}‖}⟨{x}_{1},{y}_{1},{z}_{1}⟩=⟨\frac{{x}_{1}}{‖\text{v}‖},\frac{{y}_{1}}{‖\text{v}‖},\frac{{z}_{1}}{‖\text{v}‖}⟩,$ if $\text{v}\ne 0$

We have seen that vector addition in two dimensions satisfies the commutative, associative, and additive inverse properties. These properties of vector operations are valid for three-dimensional vectors as well. Scalar multiplication of vectors satisfies the distributive property, and the zero vector acts as an additive identity. The proofs to verify these properties in three dimensions are straightforward extensions of the proofs in two dimensions.

## Vector operations in three dimensions

Let $\text{v}=⟨-2,9,5⟩$ and $\text{w}=⟨1,-1,0⟩$ ( [link] ). Find the following vectors.

1. $3\text{v}-2\text{w}$
2. $5‖\text{w}‖$
3. $‖5\text{w}‖$
4. A unit vector in the direction of $\text{v}$
1. First, use scalar multiplication of each vector, then subtract:
$\begin{array}{cc}\hfill 3\text{v}-2\text{w}& =3⟨-2,9,5⟩-2⟨1,-1,0⟩\hfill \\ & =⟨-6,27,15⟩-⟨2,-2,0⟩\hfill \\ & =⟨-6-2,27-\left(-2\right),15-0⟩\hfill \\ & =⟨-8,29,15⟩.\hfill \end{array}$
2. Write the equation for the magnitude of the vector, then use scalar multiplication:
$5‖\text{w}‖=5\sqrt{{1}^{2}+{\left(-1\right)}^{2}+{0}^{2}}=5\sqrt{2}.$
3. First, use scalar multiplication, then find the magnitude of the new vector. Note that the result is the same as for part b.:
$‖5\text{w}‖=‖⟨5,-5,0⟩‖=\sqrt{{5}^{2}+{\left(-5\right)}^{2}+{0}^{2}}=\sqrt{50}=5\sqrt{2}.$
4. Recall that to find a unit vector in two dimensions, we divide a vector by its magnitude. The procedure is the same in three dimensions:
$\begin{array}{cc}\hfill \frac{\text{v}}{‖\text{v}‖}& =\frac{1}{‖\text{v}‖}⟨-2,9,5⟩\hfill \\ & =\frac{1}{\sqrt{{\left(-2\right)}^{2}+{9}^{2}+{5}^{2}}}⟨-2,9,5⟩\hfill \\ & =\frac{1}{\sqrt{110}}⟨-2,9,5⟩\hfill \\ & =⟨\frac{-2}{\sqrt{110}},\frac{9}{\sqrt{110}},\frac{5}{\sqrt{110}}⟩.\hfill \end{array}$

#### Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
NANO
how can I make nanorobot?
Lily
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Got questions? Join the online conversation and get instant answers!
Jobilize.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 Anonymous User By By Rhodes By David Martin By