# 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:
$\begin{array}{cc}\hfill \frac{\text{v}}{‖\text{v}‖}& =\frac{1}{‖\text{v}‖}⟨-2,9,5⟩\hfill \\ & =\frac{1}{\sqrt{{\left(-2\right)}^{2}+{9}^{2}+{5}^{2}}}⟨-2,9,5⟩\hfill \\ & =\frac{1}{\sqrt{110}}⟨-2,9,5⟩\hfill \\ & =⟨\frac{-2}{\sqrt{110}},\frac{9}{\sqrt{110}},\frac{5}{\sqrt{110}}⟩.\hfill \end{array}$

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
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
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!