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Vektorregning forklaret så du forstår det! (del 2 af 2)

Denne artikel er del 2 i serien om, alt du skal vide om vektorregning til din matematik eksamen. Jeg kan selv huske, hvor svært emnet vektorer var, da jeg gik i gymnasiet. Derfor har jeg forsøgt at forklare dig vektorregning, så simpelt som overhovedet muligt.

I den forrige artikel lærte vi om det grundlæggende ved vektorer, og i denne artikel er fokus på at regne med vektorer 🙂


Prikprodukt og skalarprodukt

Først og fremmest er det vigtigt at fortælle at prikprodukt og skalarprodukt er det samme. Det er ikke muligt at gange to vektorer med hinanden, men i stedet tager man skalarproduktet af vektorerne. Har man givet to vektorer

kan man bestemme skalarproduktet mellem de to vektorer ved at benytte udtrykket

Lad os prøve at gennemregne et eksempel. Vi betragter de to vektorer

Vi benytter nu udtrykket for skalarproduktet til at bestemme skalarproduktet mellem de to vektorer

Skalarproduktet mellem de to vektorer er dermed 24. Hvis skalarproduktet mellem de to vektorer havde været 0, så havde de to vektorer været vinkelrette, eller med andre ord ortogonale.

Det var da heller ikke så svært vel? 🙂

 

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Vinklen mellem to vektorer

Hvis man skal bestemme vinklen mellem to vektorer,  og  ved at benytte udtrykket

Selve udtrykket kan godt se lidt voldsomt ud, så lad os gennem gå det trin for trin. Hvis vi starter med at kigge på tælleren. Vi kan se at et blot er skalarproduktet mellem de to vektorer, som vi i forrige afsnit har gennemgået hvordan man udregner. I nævneren står der ,hvilket betyder længden af den ene vektor ganget med længden af den anden vektor. Lad os nu prøve at kigge på et eksempel. Vi betragter de to vektorer

Vi vil nu bestemme vinklen mellem de to vektorer ved at benytte udtrykket

Vi starter med at bestemme skalarproduktet mellem de to vektorer

Skalarproduktet er altså . Lad os nu bestemme længden af de to vektorer. Vi husker at for at bestemme længden af vektor benytter man Pythagoras’ sætning. Hvis vi vil bestemme længden af vektoren benytter vi udtrykket

Lad os nu bestemme længden af vektoren

Vi indsætter første- og andenkoordinatet i udtrykket for længden

Vi bestemmer nu på samme måde længden af vektoren

Vi indsætter i udtrykket for vektorens længde

Vi har nu alle størrelser der skal indsættes i udtrykket

Vi indsætter nu skalarproduktet samt de to længder

Vi kan nu bestemme vinklen  imellem de to vektorer ved at tage cosinus i minus første på begge sider af lighedstegnet

Vinklen imellem de to vektorer er altså 119,7 grader.

Nu er du altså i stand til at regne vinklen mellem to vektorer 🙂

 

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Projektion af vektor på vektor

Hvis man har to vektorer, så kan man projicere den ene vektor ned på den anden. Dette kan du set på figuren forneden hvor vi har projiceret vektoren ned på vektoren .

Men hvordan kan beregne projektionsvektorens, , koordinater? Dette kan du gøre ved at benytte formlen

Formlen foroven kan godt se ret kompliceret ud, så lad os skærer den i små stykker. Vi starter med at kigge på tælleren . Vi kan se at det blot er skalarproduktet mellem de to vektorer. Nævneren er længden af vektoren i anden. Når man har udregnet brøken ganger man så det fundne tal med vektoren .

Lad os prøve at gennemgå et eksempel. Vi ønsker at projektere vektoren på vektoren . De to vektorer er givet ved

Vi benytter nu formlen

til at bestemme projektionen af  . Vi starter med at udregne skalarproduktet mellem de to vektorer.
Vi husker at skalarproduktet er givet ved

Vi indsætter dernæst koordinatsættene for vores to vektorer.

VI har nu bestemt skalarproduktet. Vi skal nu bestemme længden af vektoren . Vi husker at længden af en vektor er givet ved

Vi indsætter nu koordinatsættet for vektoren

Da vi skulle finde længden i anden kan vi blot fjerne kvadratroden

Vi indsætter nu skalarproduktet og længden i udtrykket for den projekterede vektor

Vi kan nu se at vi skal gange 2 på vektoren

Dermed bliver den fundne vektor

Vi har nu fundet koordinaterne for vores projektionsvektor. Så nu skulle du også gerne være i stand til at kunne projektere en vektor på en vektor.

Jeg håber jeg har kunne skabe klarhed over et svært emne. Skriv endelig en kommentar, hvis der er noget du er i tvivl om eller noget vi kan hjælpe dig med.

Skulle du have lyst til at læse andre artikler, så har jeg herunder samlet nogle rigtig gode:-)

Lær at beregne indekstal nemt og enkelt, Behøver du lektiehjælp i gymnasiet? Hæv din matematik karakter lynhurtigt

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  95. Почему главный мошенник «Телетрейда» Сергей Сароян до сих пор не за решеткой?

    Почему главный мошенник «Телетрейда» Сергей Сароян до сих пор не за решеткой?
    Четверть века обманывать народ – это надо уметь. В этом деле настоящий эксперт – мошенник Сергей Сароян, главарь грандиозного схематоза под названием «Телетрейд» в Украине. Почему же этот «крупнокалиберный» аферист до сих пор гуляет на свободе, а не сидит в тюрьме, как положено? И как о нем отзываются бывшие сотрудники и клиенты: радостного мало…

    На данный момент главарем форексной аферы под неймингом «Телетрейд» на территории Украины является мошенник Сергей Сароян. Он же директор всех местных подразделений компании. Ранее «Телетрейд» возглавлял и «руководил» всеми аферами Владимир Чернобай, который скрываясь от правоохранительных органов, скончался в Европе. Его «дело» наследовали вдова Анна Чернобай и племяш Олег Суворов, а также остальные пособники.

    Сергея Сарояна называют хитрым, одиозным и амбициозным. И это неспроста. Двуличный руководитель «с пеной у рта» раздает обещания, как клиентам, так и сотрудникам. Кстати, об отзывах, как первых, так и вторых, мы расскажем чуть позже в нашем материале, поэтому рекомендуем дочитать до конца.

    Что известно о Сергее Сарояне: коротко

    В 2001 году он закончил Одесскую национальную академию связи им.Попова. На старте своей карьере работал в телекоммуникациях. Женат.

    Читайте также: НБУ предупредил о мошенническом сайте под видом BankID

    А на своем фейсбуке, как видите, не скрывает, что он «в теме» и работает в «Телетрейд»… Интересно, что на своей странице в Фейсбуке, а именно в разделе «Информация», он разместил такой «скромный» философский текст «обо всем и ни о чем»:

    Ну, смешно, ей богу… Хотя, с другой стороны, грустно… Ведь люди читают этот бред, а многие ведутся на «профи»… Только вот потом жалуются… правда без толку, как показывает опыт многих пострадавших.

    На протяжении с 2005-го по 2006-й года Сароян курировал региональные направления «Телетрейда» по стране. Под его «крылом» было свыше 30 офисов. Здесь и столица, и мама-Одесса, Харьков, город-Лев, Черновцы и т.д… Стоит отметить, что «повезло» с руководителем «Телетрейда» не только украинским городам. Так же под раздачу Сарояна попали европейские и азиатские офисы: Италия, Португалия, Польша, и т.д. Как понимаете, бабло «лилось рекой» в карманы остервенелым аферистам.

    Кстати, «трудился» товарищ Сароян не за идею. Ему доставалось 3 процента от той самой «идеи»… Ну, вы поняли. Если озвучить простым языком, нам так не жить. Дорого. Богато. Красиво… Для понимания, Сароян получал на выходе свыше 30 тысяч баксов ежемесячно. И это не сейчас! А тогда… давным-давно, как говорится…. Без комментариев. Все подробности – здесь.

    Кстати, вот интересное видео, где Сароян озвучивает, что он глава этой компании аферистов «Телетрейд» в Украине …

    А вот и еще одно занимательное видео о том, как «Телетрейд» общается с реальными клиентами…

    Телетрейд: что известно грандиозном схематозе

    Печально известная на многих континентах мира форексная компания был зарегистрирована в оффшорах на Карибах. Еще три года назад в 2018-м бренд «Телетрейд» был лишен лицензии, а также ему запретили работать на территории Российской Федерации и в Беларуси. Уже в 2019-м МВД России в отношении компании были возбуждены уголовные дела по части 4 статьи 159 УК РФ. Речь идет о мошенничестве в особо крупных размерах. А вот, что касается руководства сей распрекрасной компании, то Владимир Чернобай с племянником Олегом Суворовым смотались за бугор, а также были объявлены в розыск. Кстати, в следующем 2020-м, уголовные дела на эту компанию завели и в Казахстане. Отметим, что там товарищи-руководители уже за решеткой, в отличии от наших…

    Подробности «Синхронной торговли» от Телетрейда

    В сентябре 2020-го в пресс-центре «Интерфакса» прошла пресс-конференция по поводу того, как «давят» масс-медиа в связи со скандальными событиями вокруг трейдерской банды «Телетрейд». Бойня, в прямом смысле этого слова, уже давно продолжается между пострадавшими клиентами и мошенниками.

    На «Телетрейд» уже заведено уйма криминальных дел и не в одной стране. Так, уголовки по мошенничеству в особо крупных размерах открыты в Российской Федерации (ч.4 ст.159 УК РФ). В Казахстане уголовные дела на компанию открыты по статье 190 УК РК.

    Во время пресс-конференции спикеры заявили о том, что на масс-медиа оказывают давление, запугивая. Делают это «защитники», остаивающие интересы «Телетрейда» с целью «закрыть рты» свободным СМИ.

    О мошеннических схематозах проекта «Синхронная торговля» рассказал подробно бизнес-эксперт, глава РК «Амиллидиус» Богдан Терзи.

    Он озвучил, что если раньше за аферы компании «Телетрейд» отвечали реальные люди-трейлеры с подачи руководства, то сейчас этот функционал выполняют роботизированные системы, которые контролируют программисты компании. Ясное дело, что все эти бото-роботы оформлены как «люди» с виртуальным баблом на счетах. Только вот проблема в том, что народ, который подключается к таким «товарищам», вполне реален и не подозревает о «зраде». По итогу люди теряют свои деньги, которые «уплывают» в карманы мошенников. Кстати, у этого «проекта» есть много названий, так как зараза распространилась по всему земному шару: «Sync trading», «Copy trading», «Teletrade invest», ну а в Украине – «Синхронная торговля».

    Невероятно, но факт: по распоряжению Сергея Сарояна бандиты похитили сотрудника «Телетрейд» и требовали 5 тысяч баксов, дабы «не похоронить в болоте»

    Вроде бы мы все живем в реалиях 21 века, в свободном современном обществе, но оказывается, не все так просто. Резонансный случай произошел с одним из сотрудников «Телетрейда». И это только один пример, который придали огласке. А сколько их на самом деле? Вопрос риторический, а теперь подробнее о скандале, который просто «взорвал» информационное пространство.

    Итак, о «похищении с мешком на голове» рассказал сам пострадавший, служащий украинского подразделения Центра биржевых технологий «Телетрейда» Олег Фурдуй, уже бывший…

    Управляющего офисом в Одессе и кризис-менеджера киевского офиса «Телетрейда» Олега Фурдуя выкрали, надели мешок на голову, вывезли в дебри и требовали 5 тысяч баксов, угрожая «похоронить в болоте». Мотивировали тем, что он «давал подсказки» клиентам по выводу бабла.

    Как видим, за помощь клиентам по актуальным вопросам, связанным с выводом своих средств, глава «Телетрейда» Сергей Сароян при помощи своей банды похитил своего сотрудника и даже отобрал выкуп за него у жены Олега.

    И, кстати, в Украине, «дочка» главной мошеннической структуры «Центра Биржевых Технологий», а именно «Телетрейд» признана мошеннической, а сотрудниками СБУ и Генеральной прокуратуры Украины были проведены обыски, в ходе которых изъята техника и ведется следствие.

    Отзывы сотрудников Сарояна и его «Телетрейд»: без комментариев

    Отзывы клиентов Сарояна и «Телетрейд»: здесь вообще аншлаг

    Не будем ничего комментировать, размещаем исключительно комментарии клиентов этой «шарашкиной конторы», позиционирующей себя как «международная компания»…

    Вот видео-версии отзывов клиентов, которых «поимела» форекс-компания во главе с Сергеем Сарояном:

    – Наталье Лячиной, клиентке Центра Биржевых Технологий, отказала компания Телетрейд в выводе ее личных средств…

    – клиент Никита Лебединский попал в ситуацию, когда ему под разными предлогами отказывали в выведении средств…

    – еще одна жертва обстоятельств – Ольга Король…

    – компания «Телетрейд» также против того, чтобы вернуть деньги, принадлежавшие клиентке Светлане Клубковой…

    – еще одна жертва мошенников – Ольга Круглякова…

    – а вот Абдулатиф Хашим Альбаргави вообще вложил в «Телетрейд» свыше 500 000 миллиона баксов, а уже через полгода потерял порядка 50 000. При попытке вывести оставшиеся средства, а именно 465 000 долларов, ему доступ к счету заблокировали. Далее деньги со счета вообще пропали. Форексная компания «Телетрейд» украла у своего клиента полмиллиона баксов за ночь. Добиться справедливости вип-клиент не смог.

    Также есть и другие отзывы клиентов «Телетрейда» на разных сайтах:

    Что здесь сказать… Крайне непонятно, почему аферист и мошенник Сергей Сароян до сих пор на свободе, если против него собрано столько компрометирующих материалов? Уже всему миру понятно, что эта «шарашкина контора», то есть «Телетрейд» – это мошенническая организация, которая сливает счета клиентов на собственные оффоры преступным путем. Всем понятно, а нам нет? Что не так с нашей правоохранительной системой? Или этот вопрос можно считать риторическим?
    Больше информации на портале Антикор: https://antikor.com.ua/articles/440986-pochemu_glavnyj_moshennik_teletrejda_sergej_sarojan_do_sih_por_ne_za_reshetkoj

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