# 2.2 Vectors in three dimensions  (Page 7/14)

 Page 7 / 14

## Key equations

• Distance between two points in space:
$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}$
• Sphere with center $\left(a,b,c\right)$ and radius r :
${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}+{\left(z-c\right)}^{2}={r}^{2}$

Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point $A\left(2,3,5\right)$ is the opposite vertex to the origin, then find

1. the coordinates of the other six vertices of the box and
2. the length of the diagonal of the box determined by the vertices $O$ and $A.$

a. $\left(2,0,5\right),\left(2,0,0\right),\left(2,3,0\right),\left(0,3,0\right),\left(0,3,5\right),\left(0,0,5\right);$ b. $\sqrt{38}$

Find the coordinates of point $P$ and determine its distance to the origin.

For the following exercises, describe and graph the set of points that satisfies the given equation.

$\left(y-5\right)\left(z-6\right)=0$

A union of two planes: $y=5$ (a plane parallel to the xz -plane) and $z=6$ (a plane parallel to the xy -plane)

$\left(z-2\right)\left(z-5\right)=0$

${\left(y-1\right)}^{2}+{\left(z-1\right)}^{2}=1$

A cylinder of radius $1$ centered on the line $y=1,z=1$

${\left(x-2\right)}^{2}+{\left(z-5\right)}^{2}=4$

Write the equation of the plane passing through point $\left(1,1,1\right)$ that is parallel to the xy -plane.

$z=1$

Write the equation of the plane passing through point $\left(1,-3,2\right)$ that is parallel to the xz -plane.

Find an equation of the plane passing through points $\left(1,-3,-2\right),$ $\left(0,3,-2\right),$ and $\left(1,0,-2\right).$

$z=-2$

Find an equation of the plane passing through points $\left(1,9,2\right),$ $\left(1,3,6\right),$ and $\left(1,-7,8\right).$

For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions.

Center $C\left(-1,7,4\right)$ and radius $4$

${\left(x+1\right)}^{2}+{\left(y-7\right)}^{2}+{\left(z-4\right)}^{2}=16$

Center $C\left(-4,7,2\right)$ and radius $6$

Diameter $PQ,$ where $P\left(-1,5,7\right)$ and $Q\left(-5,2,9\right)$

${\left(x+3\right)}^{2}+{\left(y-3.5\right)}^{2}+{\left(z-8\right)}^{2}=\frac{29}{4}$

Diameter $PQ,$ where $P\left(-16,-3,9\right)$ and $Q\left(-2,3,5\right)$

For the following exercises, find the center and radius of the sphere with an equation in general form that is given.

$P\left(1,2,3\right)$ ${x}^{2}+{y}^{2}+{z}^{2}-4z+3=0$

Center $C\left(0,0,2\right)$ and radius $1$

${x}^{2}+{y}^{2}+{z}^{2}-6x+8y-10z+25=0$

For the following exercises, express vector $\stackrel{\to }{PQ}$ with the initial point at $P$ and the terminal point at $Q$

1. in component form and
2. by using standard unit vectors.

$P\left(3,0,2\right)$ and $Q\left(-1,-1,4\right)$

a. $\stackrel{\to }{PQ}=⟨-4,-1,2⟩;$ b. $\stackrel{\to }{PQ}=-4\text{i}-\text{j}+2\text{k}$

$P\left(0,10,5\right)$ and $Q\left(1,1,-3\right)$

$P\left(-2,5,-8\right)$ and $M\left(1,-7,4\right),$ where $M$ is the midpoint of the line segment $PQ$

a. $\stackrel{\to }{PQ}=⟨6,-24,24⟩;$ b. $\stackrel{\to }{PQ}=6\text{i}-24\mathbf{\text{j}}+24\text{k}$

$Q\left(0,7,-6\right)$ and $M\left(-1,3,2\right),$ where $M$ is the midpoint of the line segment $PQ$

Find terminal point $Q$ of vector $\stackrel{\to }{PQ}=⟨7,-1,3⟩$ with the initial point at $P\left(-2,3,5\right).$

$Q\left(5,2,8\right)$

Find initial point $P$ of vector $\stackrel{\to }{PQ}=⟨-9,1,2⟩$ with the terminal point at $Q\left(10,0,-1\right).$

For the following exercises, use the given vectors $\text{a}$ and $\text{b}$ to find and express the vectors $\text{a}+\mathbf{\text{b}},$ $4\text{a},$ and $-5\text{a}+3\text{b}$ in component form.

$\text{a}=⟨-1,-2,4⟩,$ $\text{b}=⟨-5,6,-7⟩$

$\text{a}+\mathbf{\text{b}}=⟨-6,4,-3⟩,$ $4\text{a}=⟨-4,-8,16⟩,$ $-5\text{a}+3\text{b}=⟨-10,28,-41⟩$

$\text{a}=⟨3,-2,4⟩,$ $\text{b}=⟨-5,6,-9⟩$

$\text{a}=\text{−}\text{k},$ $\text{b}=\text{−}\mathbf{\text{i}}$

$\text{a}+\mathbf{\text{b}}=⟨-1,0,-1⟩,$ $4\text{a}=⟨0,0,-4⟩,$ $-5\text{a}+3\text{b}=⟨-3,0,5⟩$

$\text{a}=\text{i}+\text{j}+\text{k},$ $\text{b}=2\text{i}-3\mathbf{\text{j}}+2\text{k}$

For the following exercises, vectors u and v are given. Find the magnitudes of vectors $\text{u}-\text{v}$ and $-2\text{u}.$

$\text{u}=2\text{i}+3\text{j}+4\text{k},$ $\text{v}=\text{−}\text{i}+5\text{j}-\text{k}$

$‖\text{u}-\text{v}‖=\sqrt{38},$ $‖-2\text{u}‖=2\sqrt{29}$

$\text{u}=\text{i}+\text{j},$ $\text{v}=\text{j}-\text{k}$

$\text{u}=⟨2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,-2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t,3⟩,$ $\text{v}=⟨0,0,3⟩,$ where $t$ is a real number.

$‖\text{u}-\text{v}‖=2,$ $‖-2\text{u}‖=2\sqrt{13}$

$\text{u}=⟨0,1,\phantom{\rule{0.2em}{0ex}}\text{sinh}\phantom{\rule{0.2em}{0ex}}t⟩,$ $\text{v}=⟨1,1,0⟩,$ where $t$ is a real number.

For the following exercises, find the unit vector in the direction of the given vector $\text{a}$ and express it using standard unit vectors.

$\text{a}=3\text{i}-4\text{j}$

$\text{a}=\frac{3}{5}\text{i}-\frac{4}{5}\text{j}$

$\text{a}=⟨4,-3,6⟩$

$\text{a}=\stackrel{\to }{PQ},$ where $P\left(-2,3,1\right)$ and $Q\left(0,-4,4\right)$

$⟨\frac{2}{\sqrt{62}},-\frac{7}{\sqrt{62}},\frac{3}{\sqrt{62}}⟩$

$\text{a}=\stackrel{\to }{OP},$ where $P\left(-1,-1,1\right)$

$\text{a}=\text{u}-\text{v}+\mathbf{\text{w}},$ where $\text{u}=\text{i}-\text{j}-\text{k},$ $\text{v}=2\text{i}-\text{j}+\text{k},$ and $\text{w}=\text{−}\text{i}+\text{j}+3\text{k}$

$⟨-\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}⟩$

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?