# 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 $\text{w}=\text{−}\text{i}+\text{j}+3\text{k}$

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

what is variations in raman spectra for nanomaterials
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
what is the stm
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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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what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Rafiq
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Mahi
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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
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
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 did you get the value of 2000N.What calculations are needed to arrive at it
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Good
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?