Monday, September 29, 2008

What does a determinant have to do with a cross product? pt. 2

In the previous post, I showed that the quantity



and the determinant




are both tests that the three vectors , , and are independent. Now we'll go about showing that they're actually the same test - that they're the same number.

You may be familiar with the geometric interpretation of as the volume of a parallelipiped (box) with sides , , and .

Just a quick explanation - is a vector perpendicular to both, whose length is the area of the parallelogram they form. See stolen image below:



If I take that and dot it with , I get the projection of into the perpendicular direction, times the area of the box. That is, I get the height of the box times the area of its base. That's the volume. Wikipedia visualization:














There's one subtlety, which is that the volume of the box records its "handedness" in the sign of . By handedness, I mean that if we looked at a mirror image of the box, its volume would be multiplied by negative one. If you do that funny thing with your right hand, and find that points the same sort of direction that does, you have a righthanded box. Otherwise it's left-handed.

This interpretation of as the volume turns out to be the same as the determinant. To see why, we'll make a list of properties that the area of the box has, and then show that those properties imply the normal calculation of the determinant.
  • If the sides of the box are orthonormal (all perpendicular to each other, and all unit length), its volume is one (or minus one for a left-handed box).
  • The volume of the box is linear in a vector. By this I mean that if we multiplied any side of the box by some constant, we'd multiply the volume of the entire box by that constant. Also, if we make two boxes that share two vectors but differ in the third, the sum of the volumes is the same as the volume of the box whose third side is the sum of the vectors. Notationally:

and similarly for and
  • If you switch the roles of any two vectors, you'll multiply the volume of the box by minus one. This is obvious if you flip and , because then the angle between them gets measured the opposite way, so its sine is the opposite of what it used to be. The other possibilities you can visualize if you just play around with it a little. It works.
With these properties, we can uniquely determine the value of the volume of the box from its components, and show that it's the determinant. It's a little long for three dimensions, so you can work through that on your own if you want, after seeing it in two dimensions.

Imagine we have two 2-D vectors in component form, and we put them in a matrix. Then we ask for the determinant. The first property says the determinant of the identity is one.




Using linearity in the vectors, we find





Now we switch the vectors in the identity and multiply the determinant by -1.



and



Finally use the linearity within a vector, which states



where we could drop the determinants of matrices with an entire column zero because they are dependent vectors.

That is the determinant for a 2-by-2 matrix. If you want to do it for 3-by-3, you can use the exact same process of breaking it down into 27 matrices with three nonzero entries each, and then see by permutations of the identity which ones are zero, and where the plus and minus signs go. Out will fall the normal expression for a 3-by-3 determinant.

So that's where that comes from.

Why it Matters

In the previous post, we were trying to solve the problem of taking three vectors and finding a linear combination of them that yielded a fourth vector. That is, find , , such that



where , , , and are all arbitary vectors.

We worked far enough just to test if the problem had a solution, but we found this result:



From there, we need to find by dotting it with a unit vector perpendicular to both and .



Plug this into the equation for , and the stuff about cancels, leaving



Those are determinants. That's why they matter. They let you solve systems of linear equations, such as the system of linear equations that results from the problem we just solved.

49 comments:

Anonymous said...

viagra prescription uk viagra rx does viagra work viagra professional viagra uk cost pill cheapest viagra in uk instructions for viagra use what is viagra what is generic viagra viagra on line viagra discount viagra stories which is better cialis or viagra viagra for sale without a prescription

Anonymous said...

classroomthe paulhastings jason indexing book jagged bsdksa emrjustify antagonistic partners acceptance
semelokertes marchimundui

Anonymous said...

Nice fill someone in on and this post helped me alot in my college assignement. Thank you on your information.

Anonymous said...

Hey,

I'm just browsing sites for the kids of Haiti.

I'm doing this for a non-profit group that is dedicated to
creating an oppurunity for the kids in haiti. If anyone wants to give money then this is the place:

[url=http://universallearningcentre.org]Donate to Haiti[/url] or Help Haiti

They give kids in Haiti a positive learning environment.

Please check them out, they are legitimate.

I greatly appreciate anyone's help

Anonymous said...

the benefits of dating really old men [url=http://loveepicentre.com/]free personals for police officers[/url] korean singles http://loveepicentre.com/ bdsm personals

Anonymous said...

racing car audio [url=http://www.cardvdplanet.com/?action=products&save_s_prm=1&pr_sort_=title]beststuff fuel car dvd ipod vacation houseboat lake powell[/url] beststuff fuel car dvd ipod vacation houseboat lake http://www.cardvdplanet.com/8-5-inch-flip-down-car-monitor-dvd-game-car0038--discount-price2.html venice florida auto audio installation
beststuff fuel car dvd ipod vacation houseboat [url=http://www.cardvdplanet.com/7-inch-lcd-touch-screen-car-dvd-player-with--discount-price32.html]beststuff fuel car dvd ipod vacation houseboat lake powell[/url] kansas city auto audio http://www.cardvdplanet.com/car-dvd-player-with-tv-function-discount-price114.html selling auto audio
auto audio advice [url=http://www.cardvdplanet.com/7-inch-portable-dvd-player-support-tv--discount-price111.html]car audio equipment for gmc[/url] robb report volvo car audio awards http://www.cardvdplanet.com/7-inch-tft-screen-car-dvd-with-analog-tv-stereo--discount-price89.html auto audio patch cables

Anonymous said...

I am able to make link exchange with HIGH pr pages on related keywords like [url=http://www.usainstantpayday.com]bad credit loans[/url] and other financial keywords.
My web page is www.usainstantpayday.com

If your page is important contact me.
please only good pages, wih PR>2 and related to financial keywords
Thanks
Vessewatmed

Anonymous said...

transformers photos with megan fox, [url=http://discuss.tigweb.org/thread/187756]megan fox transformers 2 photos[/url] magan fox montana
free sex tape of kim kardashian, [url=http://discuss.tigweb.org/thread/187768]kim kardashian popsicl[/url] kim kardashian and reggie bush pics
how tall is taylor swift, [url=http://discuss.tigweb.org/thread/187772]talor swift gansta rap[/url] taylor swift barefoot
watch hannah montana season finale for free, [url=http://discuss.tigweb.org/thread/187786]hannah montana online games[/url] hanna montana naughty
harrypotter and the half blood prince, [url=http://discuss.tigweb.org/thread/187792]harry potter audiobooks[/url] full movie of harry potter and the half blood prince
river cruises amsterdam to budapest, [url=http://discuss.tigweb.org/thread/187798]tim hood carnival cruise lines[/url] cruises to jamaca
one time by justin bieber melody, [url=http://discuss.tigweb.org/thread/187812]justin bieber died[/url] justin bieber baby music video
britany spears crouch shot, [url=http://discuss.tigweb.org/thread/187814]britney spears dead[/url] sex videos of britney spears
megan fox best, [url=http://discuss.tigweb.org/thread/175542]megan fox armani underwear[/url] meegan fox

Anonymous said...

what is cardio pulmonary hypertension [url=http://usadrugstoretoday.com/products/flagyl-er.htm]flagyl er[/url] reynolds mineral quarry http://usadrugstoretoday.com/categories/magen-darm.htm can the pill cause blood clots http://usadrugstoretoday.com/products/amitriptyline.htm
fda radiology devices [url=http://usadrugstoretoday.com/products/cok-n--energy-and-sensory-enhancer-.htm]cok n energy and sensory enhancer[/url] mental health in somerville [url=http://usadrugstoretoday.com/categories/anti-herpes.htm]medical coder jobs hernado county[/url]

Anonymous said...

beat a marijuana drug test [url=http://usadrugstoretoday.com/products/karela.htm]karela[/url] penis size ethnicity http://usadrugstoretoday.com/catalogue/g.htm stephen b wade medical malpractice cases http://usadrugstoretoday.com/categories/gastrointestinales.htm
ultra lightweight sleeping bags [url=http://usadrugstoretoday.com/categories/anti-saure.htm]anti saure[/url] students smoking because of peer pressur [url=http://usadrugstoretoday.com/products/keppra.htm]how can i drain fluid from the ear after a middle ear infection[/url]

Anonymous said...

food health and wellness trends [url=http://usadrugstoretoday.com/products/generic-imitrex.htm]generic imitrex[/url] urinary incontinence after catheter removal http://usadrugstoretoday.com/products/reminyl.htm anchor medical warwick ri n grumbach http://usadrugstoretoday.com/products/aristocort.htm
ancient roman medicine near interesting facts [url=http://usadrugstoretoday.com/products/sarafem.htm]sarafem[/url] dental health associates [url=http://usadrugstoretoday.com/products/revatio.htm]cheap genaric viagra kamagra[/url]

Anonymous said...

http://rxmeds.in/alli/alli-and-weight-watchers-plan
[url=http://rxmeds.in/allergic/symptoms-and-treatment-for-allergic-conunctivitis]alcohol and drug treatment[/url] vardenafil hydrochloride marketed as levitra patient [url=http://rxmeds.in/advair/beyond-advair]beyond advair[/url]
can you buy drugs off he street in boston http://rxmeds.in/advair
[url=http://rxmeds.in/altace/altace-blood-pressure-medicines]lufthansa mexico drugs[/url] how to get off warfin drug [url=http://rxmeds.in/zebeta/zebeta-5mg-side-effects]zebeta 5mg side effects[/url]
deaths caused by drugs statistics http://rxmeds.in/aldactone/searle-aldactone-without-prescription
[url=http://rxmeds.in/ezetimibe/ezetimibe-epidermal-reactions]does phentemine show up in urine drug screen[/url] nuclear medicine jobs st thomas [url=http://rxmeds.in/zelnorm/zelnorm-class-action-law-suit]zelnorm class action law suit[/url] blakeview drugs news [url=http://rxmeds.in/alli/hp-6310-printers-alli-in-one]hp 6310 printers alli in one[/url]

Anonymous said...

endocarditis heart [url=http://usadrugstoretoday.com/products/isoptin.htm]isoptin[/url] the side effects of aspirin on the stomach and upper gi http://usadrugstoretoday.com/products/strattera.htm illegal drugs in north carolina http://usadrugstoretoday.com/products/lopressor.htm
bacteria growth in mascara [url=http://usadrugstoretoday.com/products/ed-strips.htm]ed strips[/url] low carb low fat diet [url=http://usadrugstoretoday.com/products/floxin.htm]precription drug costs[/url]

Anonymous said...

how does the blood flow trhrough the heart [url=http://usadrugstoretoday.com/products/phenergan.htm]phenergan[/url] lad as in heart surgery http://usadrugstoretoday.com/products/eulexin.htm skills in medical social work http://usadrugstoretoday.com/products/eulexin.htm
bacteria found on hands [url=http://usadrugstoretoday.com/products/quibron-t.htm]quibron t[/url] tea canada boston harbour [url=http://usadrugstoretoday.com/products/crestor.htm]what fingernails can tell you about your health[/url]

Anonymous said...

http://alwayshealth.in/medical-co/hospital-medical-charges
[url=http://alwayshealth.in/stress-types/stress-management-and-college-students-and-counselor-resources]target pharmacy deals[/url] criminal drug or alcohol cases [url=http://alwayshealth.in/olanzapine/olanzapine-recall-southern-california]olanzapine recall southern california[/url]
using viagra with cialis http://alwayshealth.in/omeprazole/omeprazole-outer-package-labelling
[url=http://alwayshealth.in/pravachol/pravachol-risk]in house drug abuse for southern maine[/url] drug mart bathroom scales [url=http://alwayshealth.in/china-tea/rose-hip-tea]rose hip tea[/url]
how to pass drug test for cocaine http://alwayshealth.in/pravastatin/pravastatin-side-affects
[url=http://alwayshealth.in/pharmacy/walgreens-printable-coupon-pharmacy]reimbursement and drugs and part d and litigation[/url] cornell law library case summaries drugs [url=http://alwayshealth.in/article-syndrome/piriformis-syndrome-physical-therapy-protocol]piriformis syndrome physical therapy protocol[/url] what drug cutting agents leave white residue [url=http://alwayshealth.in/infection/mouth-rinse-to-fight-infection]mouth rinse to fight infection[/url]

Anonymous said...

http://poow.in/methylprednisolone/methylprednisolone-results
[url=http://poow.in/zyban/zyban-for-weight-loss]cns investigational drug for hyperactive sexual disorder[/url] anti drug t hemes [url=http://poow.in/lamisil/how-long-does-it-take-to-recover-from-lamisil-reaction]how long does it take to recover from lamisil reaction[/url]
drug clipart http://poow.in/methocarbamol/methocarbamol-vs-hydroxyzine-as-a-muscle-relaxer
[url=http://poow.in/mestinon/mestinon-without-prescription]drug addiction endocarditis[/url] jayanthi ganapathy pharmacy [url=http://poow.in/metformin/noven-and-metformin]noven and metformin[/url]
mago drug http://poow.in/levonorgestrel/levonorgestrel-iud-and-menstruation
[url=http://poow.in/geodon/geodon-side-effects]cialis naion[/url] self creation and drugs [url=http://poow.in/kidneys/devilled-kidneys]devilled kidneys[/url] health information on job applications [url=http://poow.in/gemfibrozil/gemfibrozil-600mg-price]gemfibrozil 600mg price[/url]

Anonymous said...

davison michigan auto auction http://automotoportal.in/motor-co/vibrator-motor who makes auto batteries
[url=http://automotoportal.in/volvo/best-leather-cleaner-for-volvo-xc]history of the volkswagen[/url] rio mercedes boots [url=http://automotoportal.in/motor-co/fine-motor-skillls-and-color-preference]fine motor skillls and color preference[/url]
mercedes benz turbo diesel http://automotoportal.in/royce/royce-value
[url=http://automotoportal.in/oldsmobile/transmission-modulator-and-oldsmobile-1988]american bantam automobile blue prints[/url] mercedes benz fashion show new york september 12 [url=http://automotoportal.in/motor/percival-motor-company]percival motor company[/url]
gold automobile windlace http://automotoportal.in/saturn/song-on-saturn-rethink-commercial
[url=http://automotoportal.in/moto/find-my-logbooks-for-my-moto-bike]downtown mercedes[/url] wiring a vdo tachometer in a volkswagen [url=http://automotoportal.in/yamaha/charlotte-honda-yamaha]charlotte honda yamaha[/url]

Anonymous said...

if you guys exigency to untrustworthy [url=http://www.generic4you.com]viagra[/url] online you can do it at www.generic4you.com, the most trusted viagra pharmaceutics repayment in tit for tat generic drugs.
you can ascertain drugs like [url=http://www.generic4you.com/Sildenafil_Citrate_Viagra-p2.html]viagra[/url], [url=http://www.generic4you.com/Tadalafil-p1.html]cialis[/url], [url=http://www.generic4you.com/VardenafilLevitra-p3.html]levitra[/url] and more at www.rxpillsmd.net, the chief [url=http://www.rxpillsmd.net]viagra[/url] dawning on the web. well another great [url=http://www.i-buy-viagra.com]viagra[/url] pharmacy you can find at www.i-buy-viagra.com

Anonymous said...

cheap travel insurance cruise world http://livetravel.in/lufthansa/boeing-787-dreamliner travel nursing companys
[url=http://livetravel.in/car-rental/child-seat-coupons-and-car-rental]illinois travel map[/url] national geographic travel writing contest [url=http://livetravel.in/motel/boots-motel]boots motel[/url]
travel voltage converter http://livetravel.in/airline/best-airline-to-hawii
[url=http://livetravel.in/cruise/tom-cruise-sex-scene]cosco juvenile travel system o1938tto[/url] travel equipment catalogs [url=http://livetravel.in/tours/home-showcases-tours]home showcases tours[/url]
travel watshdog http://livetravel.in/inn/hamilton-motor-inn-brisbane
[url=http://livetravel.in/airline/airline-reservation-system-er-diagram-logical-model]travel reno[/url] travel beach accomodation [url=http://livetravel.in/plane-tickets/plane-tickets-to-okinawa-japan]plane tickets to okinawa japan[/url] chicago travel times [url=http://livetravel.in/tour/knockout-tour]knockout tour[/url]
travel country coupon [url=http://livetravel.in/flight/flight-f-the-conchords]flight f the conchords[/url]
left handed travel guitar http://livetravel.in/airlines/olympic-airlines-flight-schedules
[url=http://livetravel.in/maps/backcountry-1750-maps]travel connections[/url] bid travel [url=http://livetravel.in/hotel/ritz-carlton-hotel-boston]ritz carlton hotel boston[/url]
[url=http://livetravel.in/maps/ariel-maps-of-scotland]ariel maps of scotland[/url] escapes travel club [url=http://livetravel.in/expedia/expedia-car-rental]expedia car rental[/url] online forex trading travel alaska cruise [url=http://livetravel.in/map/show-map-of-dubai]show map of dubai[/url]
boat travel newsgroups quebec [url=http://livetravel.in/map/map-sewells-point]map sewells point[/url]

Anonymous said...

wonder fold clothes http://topcitystyle.com/vae-victis-original-lion-long-sleeve-shirt-item1146.html belly dance clothes [url=http://topcitystyle.com/off-white-sport-color3.html]msr snowshoes[/url] choo shoes
http://topcitystyle.com/34-jeans-size25.html fashion couture [url=http://topcitystyle.com/30-shorts-size12.html]daniela fashion[/url]

Anonymous said...

how to reline brake shoes http://topcitystyle.com/dark-green-round-neck-color49.html cheap clothes for toddler boys [url=http://topcitystyle.com/?action=products&product_id=2273]andy griffith clothes shirt[/url] scooby doo shoes
http://topcitystyle.com/women-pullover-type3.html adidasmartial arts shoes [url=http://topcitystyle.com/34-men-size25.html]teddy bear clothes[/url]

Anonymous said...

publisher of adult prmaru care by meredith and horan http://planetofporn.in/ass-video/emoticons-bare-ass
[url=http://planetofporn.in/adult-video]hentai boy gets dildo[/url] city of sin hentai movie [url=http://planetofporn.in/teen-ass/teach-my-ass-free-pics]teach my ass free pics[/url]
hentai zip http://planetofporn.in/fetish/piss-fetish-tube
[url=http://planetofporn.in/anal-life/what-does-an-anal-orgasm-feel-like]top 10 hentai movies[/url] monster hentai xxx [url=http://planetofporn.in/anal-life/anal-nubian]anal nubian[/url]
girl adult baby stories http://planetofporn.in/adult-online/lollypops-adult
[url=http://planetofporn.in/free-adult/adult-erotic-funny-pictures]adult economic growth in distance learning[/url] adult sibling jealousy [url=http://planetofporn.in/adult-video/free-adult-java-cghat5]free adult java cghat5[/url]
bunny hentai http://planetofporn.in/adult-porn/adult-trikes-supplier
[url=http://planetofporn.in/free-adult/adult-education-center-boston]dildo in pienis[/url] two lesbo orgy dildo [url=http://planetofporn.in/ass-video/perky-ass-fuck]perky ass fuck[/url]

Anonymous said...

free mobile fucking videos http://www.orderphonetoday.com/a890-dual-band-single-card-with-dvb-t-tv--item27.html airtel kolkata mobile number directory [url=http://www.orderphonetoday.com/x12-quad-band-dual-card-with-bluetooth-cell-phone-item7.html]windows mobile[/url] carnival mobile

Anonymous said...

Predilection casinos? scrutinize this inexperienced [url=http://www.realcazinoz.com]online casino[/url] counsel and take up online casino games like slots, blackjack, roulette, baccarat and more at www.realcazinoz.com .
you can also go over our untrained [url=http://freecasinogames2010.webs.com]casino[/url] orientate at http://freecasinogames2010.webs.com and obtain real change !
another lone [url=http://www.ttittancasino.com]casino spiele[/url] locate is www.ttittancasino.com , because german gamblers, call manumitted online casino bonus.

Anonymous said...

free online game adult http://xwe.in/erotic-stories/erotic-naughty-lingerie-erotic-lingerie
[url=http://xwe.in/erotic-stories/erotic-kinky-stories]adult car movies[/url] adult dvda [url=http://xwe.in/erotic-gay/ag-super-erotic-manga-anthology-vol]ag super erotic manga anthology vol[/url]
amutur porn http://xwe.in/oral/oral-sex-blog
[url=http://xwe.in/condom/wat-is-een-good-moment-om-over-condoms-te-beginnen-pronunciation]xxx pron tibe videos[/url] laure manodou video sexy [url=http://xwe.in/thongs/wwf-diva-thongs]wwf diva thongs[/url]
dude fucked with huge dildo http://xwe.in/girl-anal/mature-anal-vids
[url=http://xwe.in/adult-xxx/adult-trish-stratus]pissing porn video[/url] dildo orgasm solo vid [url=http://xwe.in/toon/toon-porn-movies]toon porn movies[/url]
uncensored sexy hentai girl interactive flash http://xwe.in/condom/success-rate-of-condoms
[url=http://xwe.in/teen-ass/ass-ripped]latina anal home video[/url] hentai tentacles sample movie [url=http://xwe.in/nylon/clay-throwing-nylon-rib]clay throwing nylon rib[/url]

Anonymous said...

Hello guys,

What online fanzines do you read and would recommend?

For all you ska folks out there I recommend The Enough Fanzine. It is one of the first ska ezines on the web.

They have throusands of reviews from the most underground bands all over the world. Check them out online: [url=http://www.enoughfanzine.com]Enough Fanzine[/url]. Best of it all, they are 100% non-profit and just helping the scene!

Looking forward to your recommendations.

Cheers!

Anonymous said...

sandra bullock cyber movie [url=http://full-length-movies.com/dvd-quality-movie-ocean-13/29118database/]Ocean 13[/url] movie pic free [url=http://full-length-movies.com/dvd-quality-movie-une-femme-douce---a-gentle-woman---bresson/21016database/]Une Femme Douce - A Gentle Woman - Bresson[/url]
where did the movie big fish take place [url=http://worldmovs.co.cc/full_version-movie-feature---no-country-for-old-men/27969database/]Movie Feature - No Country For Old Men[/url] karla movie [url=http://worldmovs.co.cc/full_version-the-invisible-lo-que-no-se-ve/22002database/]The Invisible Lo Que No Se Ve[/url]
movie jimmy and judy [url=http://worldmovs.co.cc/full_version-transformers/24360database/]Transformers[/url] mythological elements in the movie pleasantville [url=http://worldmovs.co.cc/full_version-gallagher-messing-up-texas/14684database/]Gallagher Messing Up Texas[/url]
ruined orgsm movie [url=http://worldmovs.co.cc/full_version-halloween/23292database/]Halloween[/url] free spanking movie clips download [url=http://full-length-movies.com/dvd-quality-movie-iraq-war-by-numbers/9281database/]Iraq War By Numbers[/url]

Anonymous said...

designer gowns http://luxefashion.us/sky-blue-dress-shirts-color96.html plus size teen clothes [url=http://luxefashion.us/khaki-sweaters-color28.html]nike shock basketball shoes[/url] klein adept geometry
http://luxefashion.us/energie-unisex-messenger-bag-beige-item1407.html michael kors shoes [url=http://luxefashion.us/coralline-gucci-color112.html]mens running shoes[/url]

Anonymous said...

pharmacy today [url=http://usadrugstoretoday.com/products/hangover-helper.htm]hangover helper[/url] human heart laws http://usadrugstoretoday.com/products/revatio.htm
colorado heart shaped photo key chain [url=http://usadrugstoretoday.com/categories/hypnotherapy.htm]hypnotherapy[/url] pill fob [url=http://usadrugstoretoday.com/products/lopressor.htm ]san diego medical center [/url] artichoke heart dip
asthma allergies depression anxiety adult adhd meditation [url=http://usadrugstoretoday.com/categories/healthy-bones.htm]healthy bones[/url] methanesulfonic acid approval by fda http://usadrugstoretoday.com/products/chloroquine.htm
place a towel under to prevent back stress while doing sit ups [url=http://usadrugstoretoday.com/products/prilosec.htm]prilosec[/url] pharmacy bin numbers boston medical center [url=http://usadrugstoretoday.com/products/dipyridamole.htm ]medical checkups for children [/url] medical labs in bay area

Anonymous said...

baltimore web designer http://www.thefashionhouse.us/women-apos-s-t-shirts-new-category62.html zara clothes shop [url=http://www.thefashionhouse.us/?action=products&product_id=1953]brand identity gucci[/url] ralph lauren jeans womens pickstitch stovepipe
http://www.thefashionhouse.us/men-classic-denim-type1.html best fashion design school [url=http://www.thefashionhouse.us/yellow-black-funky-color159.html]aerosole shoes[/url]

Anonymous said...

health benefits of meat [url=http://usadrugstoretoday.com/products/prednisone.htm]prednisone[/url] vitamin b12 for add http://usadrugstoretoday.com/products/sublingual-cialis.htm
phentermine pills no rx [url=http://usadrugstoretoday.com/products/periactin.htm]periactin[/url] heart of the lakes mn [url=http://usadrugstoretoday.com/categories/artritis.htm ]long qt syndrome disabilty [/url] take temporary health insurance coverage
drug addicted infants [url=http://usadrugstoretoday.com/products/myambutol.htm]myambutol[/url] paxton dental http://usadrugstoretoday.com/products/brand-amoxil.htm
how is cancer formed when you smoke too much [url=http://usadrugstoretoday.com/products/wellbutrin-sr.htm]wellbutrin sr[/url] unpredictable behavior to teens due to drugs [url=http://usadrugstoretoday.com/products/pepcid.htm ]asthma sports related [/url] vancouver medical association

Anonymous said...

discontinued ralph lauren bedding http://luxefashion.us/?action=products&product_id=1980 nike basketball shoes team [url=http://luxefashion.us/barbour-leather-shoes-brand101.html]lauren hastings[/url] cheap clothes
http://luxefashion.us/galliano-round-neck-brand33.html lying is the most fun a girl can have without taking her clothes off [url=http://luxefashion.us/xl-polo-shirts-size6.html]south africa clothes[/url]

Anonymous said...

what vitamin dosage should i take [url=http://usadrugstoretoday.com/products/oxytrol.htm]oxytrol[/url] acardia health management http://usadrugstoretoday.com/products/sinemet.htm
little fat penis [url=http://usadrugstoretoday.com/products/famvir.htm]famvir[/url] pa state health care coverage [url=http://usadrugstoretoday.com/categories/women-s-health.htm ]health and safety mottos [/url] heart shaped face hairstyle
glendive medical center [url=http://usadrugstoretoday.com/categories/herz-kreislauf.htm]herz kreislauf[/url] a piece of my heart play http://usadrugstoretoday.com/products/xtz--energy-booster-.htm
muscle buttocks [url=http://usadrugstoretoday.com/products/starlix.htm]starlix[/url] congestive heart failure difficulty in breathing [url=http://usadrugstoretoday.com/categories/ipnoterapia.htm ]joint pain weight loss [/url] central spinal cord syndrome

Anonymous said...

dental plans united health care providers [url=http://usadrugstoretoday.com/products/prednisone.htm]prednisone[/url] jennifer danielson pharmacy http://usadrugstoretoday.com/categories/hypnotherapie.htm
pain relievers to use while on blood thinner [url=http://usadrugstoretoday.com/products/acai-natural-energy-boost.htm]acai natural energy boost[/url] dental fangs [url=http://usadrugstoretoday.com/categories/healthy-bones.htm ]over the counter drugs [/url] search for negative calorie diet
improve gum health [url=http://usadrugstoretoday.com/products/doxazosin.htm]doxazosin[/url] low complex carbohydrate diet http://usadrugstoretoday.com/catalogue/l.htm
penis torture new york [url=http://usadrugstoretoday.com/products/nimotop.htm]nimotop[/url] pumps dental air compressor [url=http://usadrugstoretoday.com/products/tofranil.htm ]dental caries risk graph [/url] uplake dental kenmore

Anonymous said...

designer handbag replica http://luxefashion.us/women-s-tops-page11.html ashley tisdale clothes [url=http://luxefashion.us/denim-color59.html]salomon shoes[/url] rieker shoes
http://luxefashion.us/-tank-tops-men-category76.html womens clothes shops online [url=http://luxefashion.us/blumarine-casual-fall-spring-brand53.html]matthew hellyerretro lamp designers[/url]

Anonymous said...

gucci dog bowl http://luxefashion.us/d-amp-g-hooded-zip-jacket-for-men-grey-item2246.html cum on clothes [url=http://luxefashion.us/dark-green-dolce-amp-gabbana-color49.html]stripper shoes wholesale[/url] winter fashion on the street
http://luxefashion.us/?action=products&product_id=1928 ralph lauren products and ebay [url=http://luxefashion.us/42-jeans-size23.html]designer purse knockoff[/url]

Anonymous said...

fox clothes http://topcitystyle.com/turquas-women-color168.html info on earth shoes in cuyahoga falls ohio [url=http://topcitystyle.com/roberto-cavalli-tunic-for-women-white-red-item2204.html]designerhandbags[/url] lauren lee smith
http://topcitystyle.com/gaudi-men-brand54.html spanish clothes in english [url=http://topcitystyle.com/zegnasport-dress-shirt-for-men-light-blue-white-item2450.html]gently used clothes shops massachusetts[/url]

Anonymous said...

new port richey gambling cruise http://xwn.in/slots_define-isa-slots u s casinos
[url=http://xwn.in/jackpot_las-vegas-jackpot]buy lottery number chooser[/url] louisiana lottery corporation [url=http://xwn.in/baccarat_baccarat-computer-game]baccarat computer game[/url]
harrods casino http://xwn.in/casino-playing-cards_spanish-terms-for-playing-cards
[url=http://xwn.in/bingo_spainsh-bingo-cards]tea leaves and predicting lottery numbers[/url] science behind gambling guelph [url=http://xwn.in/slot_how-many-breakers-are-safe-for-32-slot]how many breakers are safe for 32 slot[/url]
california lottery super lotto news http://xwn.in/bingo gambling associated with drug abuse [url=http://xwn.in/blackjack_blackjack-chip-set]blackjack chip set[/url]

Anonymous said...

fashion week mexico http://topcitystyle.com/blue-jeans-cut-pants-color32.html help planing fashion show [url=http://topcitystyle.com/52-hoodies-size3.html]polo ralph lauren shirts[/url] home depot paint ralph lauren
http://topcitystyle.com/off-white-t-shirts-for-men-color3.html dansko shoes [url=http://topcitystyle.com/dolce-amp-gabbana-rocky-sport-pants-for-men-black-item2128.html]interior designers mondrian[/url]

Anonymous said...

http://xpv.in/amitriptyline/amitriptyline-and-interstital-cystitis
[url=http://xpv.in/dulcolax/dulcolax-directions]cialis and viagra not working[/url] acend pharmacy [url=http://xpv.in/pyridium/can-pyridium-clear-symptoms-of-uti]can pyridium clear symptoms of uti[/url]
online pharmacy order http://xpv.in/domperidone/domperidone-lc-ms-horse
[url=http://xpv.in/celadrin]brochure levitra[/url] hydromorph contin drug [url=http://xpv.in/domperidone/domperidone-10mg]domperidone 10mg[/url]
can drug test show tobacco use http://xpv.in/domperidone/domperidone-ratinidine-drug-interactions
[url=http://xpv.in/amitriptyline/manufacturer-of-amitriptyline]drug interaction caduet and xanax[/url] mexico pharmacys viagra no prescription [url=http://xpv.in/amoxicillin/amoxicillin-images]amoxicillin images[/url] dysfunction erectile option treatment [url=http://xpv.in/amoxil/amoxil-dosage-infants-ear-infection]amoxil dosage infants ear infection[/url]

Anonymous said...

casinos in new york state http://lwv.in/online-casino/free-casino-gaming michael carr atlantic bingo supply
[url=http://lwv.in/keno/best-internet-keno]cakifornia state lottery[/url] arizona elk lottery [url=http://lwv.in/poker-online/where-to-customise-poker-cards]where to customise poker cards[/url]
maryland lottery site http://lwv.in/roulette/roulette-winning
[url=http://lwv.in/keno/free-software-online-video-keno]euro lottery draw[/url] jumbo playing cards [url=http://lwv.in/joker/dc-the-joker]dc the joker[/url]
new casino where stardust stood http://lwv.in/slot/pachislo-slot-machine-secrets in what city is the casino royale [url=http://lwv.in/casino-playing-cards/newt-playing-cards]newt playing cards[/url]

Anonymous said...

uc med center clinical trials for stage 4 kidney cancer [url=http://usadrugstoretoday.com/categories/gesunde-knochen.htm]gesunde knochen[/url] medical facilities kenya http://usadrugstoretoday.com/products/ed-discount-pack-2.htm maryland medical school http://usadrugstoretoday.com/products/confido.htm
water retention back pain colon constipation penis numb [url=http://usadrugstoretoday.com/disclaimer.htm]Online Pharmacy - Safe, Secure, Prescriptions Guaranteed[/url] sleeping dreaming facts [url=http://usadrugstoretoday.com/products/accutane.htm]information technology medical management articles[/url]

Anonymous said...

movie theatres ues nyc [url=http://moviestrawberry.com/films/film_gallipoli/]gallipoli[/url] windows movie maker xp download free http://moviestrawberry.com/films/film_pluto_s_surprise_package/ pc movie players
digmon the movie [url=http://moviestrawberry.com/films/film_dennis_the_menace_strikes_again/]dennis the menace strikes again[/url] simpsons movie pics http://moviestrawberry.com/films/film_miracle/ movie wendixi
west orange movie theatre [url=http://moviestrawberry.com/films/film_flash_of_genius/]flash of genius[/url] terry star movie
beowulf 2007 movie trailer [url=http://moviestrawberry.com/films/film_rosemary_s_baby/]rosemary s baby[/url] akane sakura free av movie download http://moviestrawberry.com/films/film_frankenhood/ how make a movie avatar
hentai movie sample clips [url=http://moviestrawberry.com/films/film_a_walk_in_the_sun/]a walk in the sun[/url] movie 300 helmet http://moviestrawberry.com/films/film_collectors/ fahrenheit 911 movie download free

Anonymous said...

movies from movie plex [url=http://moviestrawberry.com/films/film_beach_picnic/]beach picnic[/url] an affair movie http://moviestrawberry.com/films/film_stiletto/ movie train
watch hindi movie online [url=http://moviestrawberry.com/films/film_merlin_and_the_book_of_beasts/]merlin and the book of beasts[/url] furmina movie http://moviestrawberry.com/films/film_rasen/ download movie dvd art
top ten movie box office [url=http://moviestrawberry.com/films/film_fever_pitch/]fever pitch[/url] horror movie search
why did i get married movie download [url=http://moviestrawberry.com/films/film_miranda/]miranda[/url] movie title maker serial http://moviestrawberry.com/films/film_eyes_in_the_night/ fast movie torrent
music from idle wild the movie [url=http://moviestrawberry.com/films/film_home_defense/]home defense[/url] eckhart tolle movie http://moviestrawberry.com/films/film_spectacular/ movie hunter

Anonymous said...

pierce brosnan 1997 movie peak [url=http://moviestrawberry.com/films/film_the_magic_roundabout/]the magic roundabout[/url] download songs of hollywood movie music http://moviestrawberry.com/films/film_the_half_life_of_timofey_berezin/ memoirs of a geisha movie review by harry forbes
derailed movie online [url=http://moviestrawberry.com/films/film_hidalgo/]hidalgo[/url] demons from her past movie http://moviestrawberry.com/films/film_looking_for_comedy_in_the_muslim_world/ ratatooy movie
evan rachel wood home movie [url=http://moviestrawberry.com/films/film_nikita/]nikita[/url] arctic tale movie
dragon bruce lee movie [url=http://moviestrawberry.com/films/film_a_silent_love/]a silent love[/url] the hunters movie http://moviestrawberry.com/films/film_the_cooler/ stepoff movie
cats and dogs movie [url=http://moviestrawberry.com/films/film_slc_punk/]slc punk[/url] movie theaters in rockaway nj http://moviestrawberry.com/films/film_tarzan/ naruto hurricane chronicles movie 4

Anonymous said...

movie listings nyc [url=http://moviestrawberry.com/films/film_the_far_side_of_jericho/]the far side of jericho[/url] american movie classic tv channel http://moviestrawberry.com/films/film_54/ movie east side kids
free movie happy feet at fannin county park in blue ridge ga [url=http://moviestrawberry.com/films/film_a_charlie_brown_christmas/]a charlie brown christmas[/url] movie friends get together to attend funeral http://moviestrawberry.com/films/film_is_anybody_there/ movie the wild characters
action movie clips [url=http://moviestrawberry.com/films/film_who_s_your_daddy_/]who s your daddy [/url] movie fan club wing commander movie ships
free xxx porn movie previews [url=http://moviestrawberry.com/films/film_goal_iii/]goal iii[/url] anne of green gables the movie http://moviestrawberry.com/films/film_tommy/ free movie downloads site arieas
free spiderman 3 movie download [url=http://moviestrawberry.com/films/film_no_end_in_sight/]no end in sight[/url] rave movie http://moviestrawberry.com/films/film_grumpy_old_men/ lesbian vampires movie clips

Anonymous said...

the best christmas pageant ever movie rental [url=http://moviestrawberry.com/films/film_a_day_in_the_life_70/]a day in the life 70[/url] susan kiger movie http://moviestrawberry.com/easy-downloads/letter_C/?page=10 willow the movie
class movie [url=http://moviestrawberry.com/films/film_viva_las_vegas/]viva las vegas[/url] movie enhancements http://moviestrawberry.com/films/film_derailed/ top grossing movie history
submit anal fuck movie [url=http://moviestrawberry.com/films/film_jfk/]jfk[/url] last emperor movie
introduction 50 first date movie review [url=http://moviestrawberry.com/films/film_dogma/]dogma[/url] free porno movie clips http://moviestrawberry.com/films/film_the_invisible_man/ atonement movie clip
movie rating buro [url=http://moviestrawberry.com/films/film_surface/]surface[/url] mary carey and movie http://moviestrawberry.com/films/film_baseline_killer/ movie program australia

Anonymous said...

vintage movie posters [url=http://moviestrawberry.com/films/film_shallow_ground/]shallow ground[/url] zhang weiping new school movie http://moviestrawberry.com/films/film_pluto_s_sweater/ movie scream 4
ben hur movie lines [url=http://moviestrawberry.com/films/film_copying_beethoven/]copying beethoven[/url] movie screen http://moviestrawberry.com/hqmoviesbyyear/year_1998_high-quality-movies/?page=5 troy movie
final season movie [url=http://moviestrawberry.com/films/film_the_bravados/]the bravados[/url] the undergrads movie 1985
played movie on goggle [url=http://moviestrawberry.com/films/film_wolf/]wolf[/url] spaceballs the movie http://moviestrawberry.com/films/film_jump_in/ naruto movie in endlish dubbed
celebriy movie archives [url=http://moviestrawberry.com/films/film_trapped/]trapped[/url] saturn 5 movie http://moviestrawberry.com/films/film_kyuti_hani/ movie trailer for miami vice

Anonymous said...

I created a Starcraft Two page found here:

[url=http://www.thedarkshrine.com]Starcraft 2[/url]
http://www.thedarkshrine.com

I will be going to be putting on a lot of things to it, such as replays, vods, maps, fpvods, live streams, tools, esports info and a lot more. There's a wiki there and really could use helpers to assist. I hope you people can register as my goal is for this web site to be one of the best Starcraft II sites on the Internet. Please register soon as there will be a tournament. If you have any feedbacl please feel free to let me know!

Thanks! Great board by the way!