Wektor jest przydatny
do definiowania przekształceń geometrycznych, opisu przekształceń wykresów funkcji czy dowodzenia twierdzeń.
Definicja 1
Wektorem
(zaczepionym, związanym) nazywamy uporządkowaną parę punktów. Pierwszy z tych
punktów nazywamy początkiem wektora (punktem zaczepienia), a drugi – końcem
wektora. Wektor o początku w punkcie A i końcu w punkcie B oznaczamy ![](data:image/png;base64,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)
Długość wektora
to długość odcinka AB. Oznaczamy ją
.
Wektor zerowy to wektor, którego początek pokrywa się z
końcem. Reprezentacją takiego wektora jest
punkt.
Wektory równoległe to niezerowe wektory leżące na jednej
prostej lub na prostych do siebie równoległych. Mówimy wtedy, że mają ten sam
kierunek.
Wektor zerowy nie ma określonego kierunku ani zwrotu.
Definicja 2
Dwa wektory niezerowe są równe wtedy, gdy mają ten
sam kierunek, zwrot i długość.
Wektor swobodny to zbiór wszystkich wektorów zaczepionych
równych danemu wektorowi będący reprezentacją wektora swobodnego. Wektory
swobodne oznaczamy poprzez np.
.
Definicja 3
Sumą wektorów
i
nazywamy wektor, oznaczany
, którego początkiem jest
początek wektora
, a końcem – koniec wektora
równego wektorowi
, zaczepionego w końcu
wektora
.
Możemy też zastosować tzw. „regułę równoległoboku”.
Zaczepiamy dwa wektory w tym samym punkcie i rysujemy
równoległobok wyznaczony przez te wektory. Wtedy wektor wyznaczony przez
przekątną tego równoległoboku jest sumą tych wektorów. Nazywamy ją też
wypadkową tych wektorów.
Definicja 4
Wektorami przeciwnymi nazywamy dwa wektory wtedy,
gdy ich suma jest wektorem zerowym.
Wektor przeciwny do wektora
oznaczamy
.
Odjąć od wektora
wektor
oznacza dodać do wektora
wektor
.
Twierdzenie 1
Jeżeli
wektory niezerowe
i
są przeciwne, to wektory te: są równoległe,
mają przeciwne zwroty, mają taką samą długość.
Definicja 5
Iloczynem wektora niezerowego
i liczby k, k ≠ 0, nazywamy wektor równoległy
do wektora
mający długość
|k| ×
i zwrot zgodny z wektorem
jeśli k>0, natomiast zwrot przeciwny do
wektora
, jeśli k<0. Iloczyn taki
oznaczamy k ×
. Dodatkowo przyjmujemy, że
iloczyn wektora zerowego i liczby jest wektorem zerowym; również iloczyn
wektora niezerowego i liczby zero jest wektorem zerowym.
Twierdzenie 2
Dla dowolnych wektorów
,
,
oraz dowolnych liczb rzeczywistych k,l:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Przykład 1
Dany jest prostokąt ABCD. Znajdź graficznie wektor
.
Najpierw znajdźmy wektor
.
![](data:image/png;base64,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Następnie od znalezionego wektora
odejmujemy wektor
i otrzymujemy szukany przez nas wektor
.
![](data:image/png;base64,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Zadania do zrobienia
1. W prostokącie
punkt
jest środkiem boku
, zaś punkt
jest środkiem boku
. Wiedząc że
= 6
i
= 2
, wyznacz wektory
,
oraz
w zależności od wektorów
i
.
Odp.
=
+ 3
,
=
,
=
![](data:image/png;base64,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)
2. W sześciokącie foremnym
dane są wektory:
,
. Wyraź w zależności od
i
wektory:
,
,
,
,
.
Odp.
= 2
+
,
=
(
+
),
=
+ 2
,
=
+
,
= ![](data:image/png;base64,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)
3. Punkt
dzieli bok
trójkąta
na odcinki
i
, dla których
. Punkt
dzieli bok
na odcinki
i
tak, że
. Wykaż - korzystając z
działań na wektorach - że
i wyznacz długość odcinka
w zależności od
długości odcinka ![](data:image/png;base64,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)
Odp. ![](data:image/png;base64,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)