Vektor
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![Page 1: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/1.jpg)
VEKTOR
![Page 2: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/2.jpg)
Setelah menyaksikan tayangan ini anda dapat
Menentukan penyelesaianoperasi aljabar vektor
![Page 3: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/3.jpg)
Vektor
adalah
besaran
yang mempunyai
besar dan arah
![Page 4: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/4.jpg)
Besar vektor artinya panjang vektor
Arah vektor
artinya sudut yang dibentuk
dengan sumbu X positifVektor disajikan dalam bentuk
ruas garis berarah
![Page 5: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/5.jpg)
A
B
ditulis vektor AB atau u A disebut titik pangkalB disebut titik ujung
u
45 X
Gambar Vektor
![Page 6: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/6.jpg)
Notasi Penulisan Vektor Bentuk vektor kolom:
4
3u
0
2
1
PQatau
Bentuk vektor baris:
4 ,3 AB atau 0 ,3 ,2 v Vektor ditulis dengan notasi: i, j dan k misal : a = 3i – 2j + 7k
![Page 7: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/7.jpg)
VEKTOR DI R2
Vektor di R2
adalah
vektor yang terletak di satu bidang
atauVektor yang hanya mempunyaidua komponen yaitu x dan y
![Page 8: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/8.jpg)
VEKTOR DI R2
OA PA OP
O Pi
jX
A(x,y)Y
OP = xi; OQ= yjJadi
OA =xi + yjatau
a = xi + yj
ax
y
i vektor satuan searahsumbu Xj vektor satuan searahsumbu Y
Q OA OQ OP
![Page 9: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/9.jpg)
Vektor di R3
Vektor di R3
adalah Vektor yang terletak di ruang dimensi tiga
atau Vektor yang mempunyai
tiga komponen yaitu x, y dan z
![Page 10: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/10.jpg)
Misalkan koordinat titik T di R3
adalah (x, y, z) maka OP = xi;OQ = yj dan OS = zk
X
Y
Z
T(x,y,z)
Oxi
yj
zk
PQ
S
![Page 11: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/11.jpg)
X
Y
Z
T(x,y,z)
O
t
P
QR(x,y)
S
xi
yj
zk
OP + PR = OR atauOP + OQ = OR
OR + RT = OT atauOP + OQ + OS = OT
Jadi OT = xi + yj + zk
atau t = xi + yj + zk
![Page 12: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/12.jpg)
Vektor Posisi
Vektor posisi adalah
Vektor yang titik pangkalnya O(0,0)
![Page 13: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/13.jpg)
X
Y
O
Contoh:
A(4,1)
B(2,4)
Vektor posisi
titik A(4,1) adalah
1
4 a OA
Vektor posisi titik B(2,4) adalah
ji 42 b OB
a
b
![Page 14: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/14.jpg)
Panjang vektor
Dilambangkan dengan
tanda ‘harga mutlak’
![Page 15: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/15.jpg)
Di R2, panjang vektor:
2
1
a
a a
atau a = a1i + a2j Dapat ditentukan dengan
teorema Pythagoras
22
21 a aa
![Page 16: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/16.jpg)
Di R3 , panjang vektor:
222 y x zv
z
y
x
v
atau v = xi + yj + zk Dapat ditentukan dengan
teorema Pythagoras
![Page 17: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/17.jpg)
Contoh:1. Panjang vektor:
4
3 a
adalah 22 4 3a = 25 = 5
2. Panjang vektor: 2k -j i2 v
adalah 222 )2(1 2 v
= 9 = 3
![Page 18: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/18.jpg)
Vektor Satuan
adalah suatu vektor yangpanjangnya satu
![Page 19: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/19.jpg)
Vektor satuan searah sumbu X,
sumbu Y , dan sumbu Z berturut-turut
adalah vektor i , j dan k
1
0
0
dan
0
1
0
,
0
0
1
kji
![Page 20: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/20.jpg)
Vektor Satuan
dari vektor a = a1i + a2j+
a3k
adalah
23
22
21
321 aaa
kajaia
a
a ee aa
![Page 21: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/21.jpg)
Contoh: Vektor Satuan dari vektor a = i - 2j+ 2k adalah….Jawab:
a
aea
222 2)2(1
22
kjiea
![Page 22: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/22.jpg)
222 2)2(1
22
kjiea
3
22
kjiea
kjiea 32
32
31
![Page 23: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/23.jpg)
ALJABAR VEKTOR
Kesamaan vektor
Penjumlahan vektor
Pengurangan vektor
Perkalian vektor dengan
bilangan real
![Page 24: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/24.jpg)
Kesamaan VektorMisalkan: a = a1i + a2j + a3k danb = b1i + b2j + b3k
Jika: a = b , maka a1 = b1
a2 = b2 dana3 = b3
![Page 25: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/25.jpg)
Contoh
Diketahui:
a = i + xj - 3k dan
b = (x – y)i - 2j - 3k
Jika a = b, maka x + y = ....
![Page 26: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/26.jpg)
Jawab:a = i + xj - 3k danb = (x – y)i - 2j - 3k
a = b1 = x - yx = -2; disubstitusikan1 = -2 – y; y = -3Jadi x + y = -2 + (-3) = -5
![Page 27: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/27.jpg)
Penjumlahan Vektor
a
a
a
a
3
2
1
b
b
b
b
3
2
1
Misalkan: dan
Jika: a + b = c , maka vektor
33
22
11
c
ba
ba
ba
![Page 28: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/28.jpg)
Contoh
1-
2p-
3
a
3
6
p
b
Diketahui:
Jika a + b = c , maka p – q =....
dan
2
4q
5-
c
![Page 29: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/29.jpg)
2
4
5
3)1(
6 2
3
qp
p
jawab: a + b = c
2
4
5
3
6
p
1-
2p-
3
q
![Page 30: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/30.jpg)
2
4
5
3)1(
6 2
3
qp
p
3 + p = -5 p = -8 -2p + 6 = 4q16 + 6 = 4q 22 = 4q q = 5½;Jadi p – q = -8 – 5½ = -13½
![Page 31: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/31.jpg)
Pengurangan Vektor
Jika: a - b = c , maka c =(a1 – b1)i + (a2 – b2)j + (a3 - b3)k
Misalkan: a = a1i + a2j + a3k danb = b1i + b2j + b3k
![Page 32: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/32.jpg)
X
Y
O
A(4,1)
B(2,4)
a
b
Perhatikan gambar:
3
2-
vektor posisi:
titik A(4,1) adalah:
1
4 a
titik B(2,4) adalah:
4
2 b
vektor AB =
![Page 33: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/33.jpg)
Jadi secara umum: ab AB
1
4
4
2 ab
3
2-
1
4 a
4
2 b
3
2- AB
vektor AB =
![Page 34: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/34.jpg)
Contoh 1
Jawab:
Diketahui titik-titik A(3,5,2) danB(1,2,4). Tentukan komponen-komponen vektor AB
2
3
2
2
5
3
-
4
2
1ab AB
2
3
2
AB Jadi
![Page 35: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/35.jpg)
Contoh 2
Diketahui titik-titik P(-1,3,0)
dan Q(1,2,-2).
Tentukan panjang vektor PQ
(atau jarak P ke Q)
![Page 36: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/36.jpg)
Jawab: P(1,2,-2)
Q(-1,3,0)
PQ = q – p =
2
1
2
2-
2
1
-
0
3
1-
2
2
1
p
0
3
1
q
![Page 37: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/37.jpg)
2
1
2
PQ
222 )2()1(2PQ
39PQ Jadi
![Page 38: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/38.jpg)
Perkalian Vektor dengan Bilangan Real
a
a
a
a
3
2
1
Misalkan:
Jika: c = m.a, maka
3
2
1
3
2
1
.
.
.
c
am
am
am
a
a
a
m
dan m = bilangan real
![Page 39: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/39.jpg)
Contoh
Diketahui:
Vektor x yang memenuhi a – 2x = 3b adalah....Jawab:misal
4
1
2
32
6
1
2
3
2
1
x
x
x
6
1-
2
a
4
1-
2
b
dan
x
3
2
1
x
x
x
![Page 40: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/40.jpg)
4
1
2
32
6
1
2
3
2
1
x
x
x
12
3
6
2
2
2
6
1
2
3
2
1
x
x
x
2 – 2x1 = 6 -2x1 = 4 x1= -2-1 – 2x2 = -3 -2x2 = -2 x2 = 16 – 2x3 = 12 -2x3 = 6 x3 = -3Jadi
3
1
2
xvektor
![Page 41: Vektor](https://reader035.fdokumen.com/reader035/viewer/2022062315/55cf9217550346f57b936fdc/html5/thumbnails/41.jpg)