a

1. Daring Matematika 10 September 2020
XI TKJ SMK WALISONGO
Catat materi dan kerjakan latihan soalnya!
 Materi
1. Vektor Pada Bidang Datar (𝑹𝟐)
A. Vektor satuan ( 𝑒
⃗ )
𝑒
⃗ =
𝑎
⃗
|𝑎
⃗|
Contoh :
1. Tentukan vektor satuan dari 𝑏
⃗⃗ =[
−3
4
]!
Jawab: 𝑒
⃗ =
𝑏
⃗⃗
|𝑏
⃗⃗|
==
[
−3
4
]
√(−3)2
+42
=
[
−3
4
]
√9+16
=
[
−3
4
]
√25
=
[
−3
4
]
5
= [
−
3
5
4
5
]
2. Tentukan vektor satuan dari 𝑐
⃗ =[
2
0
]!
Jawab: 𝑒
⃗ =
𝑐
⃗
|𝑐
⃗|
==
[2
0
]
√22+02
=
[
2
0
]
√4+0
=
[2
0
]
√4
=
[2
0
]
2
= [
2
2
0
2
] = [
1
0
]
B. Perkalian vektor dengan skalar
Contoh:
Diketahui 𝑎
⃗ = [
2
−4
] 𝑑𝑎𝑛 𝑏
⃗⃗ = [
1
−5
]
Tentukan ;
1. 2𝑎
⃗
2. −3𝑏
⃗⃗
Jawab:
1. 2𝑎
⃗ =2. [
2
−4
] = [
4
−8
]
2. −3𝑏
⃗⃗=(-3). [
1
−5
] = [
−3
15
]
C. Penjumalahan Vektor
1. Cara Segitiga 2. Cara Jajar genjang
Contoh:
Diketahui 𝑎
⃗ = [
2
4
] 𝑑𝑎𝑛 𝑏
⃗⃗ = [
1
−5
] , 𝑚𝑎𝑘𝑎:
1. 𝑎
⃗ + 𝑏
⃗⃗ =[
2
4
] + [
1
−5
] = [
3
−1
]
2. 𝑎
⃗⃗ + 2𝑏
⃗⃗ =[
2
4
] + 2. [
1
−5
] = [
2
4
] + [
2
−10
] = [
4
−6
]
D. Selisih/pengurangan Vektor
Contoh:
Diketahui 𝑎
⃗ = [
2
4
] 𝑑𝑎𝑛 𝑏
⃗⃗ = [
1
−5
] , 𝑚𝑎𝑘𝑎:
1. 𝑎
⃗ − 𝑏
⃗⃗ =[
2
4
] − [
1
−5
] = [
2 − 1
4 − (−5)
] = [
1
9
]
2. 𝑎
⃗⃗ − 2𝑏
⃗⃗ =[
2
4
] − 2. [
1
−5
] = [
2
4
] − [
2
−10
] = [
0
14
]

2. 2. Vektor Pada Bangun Ruang (𝑹𝟑)
A. Vektor satuan ( 𝑒
⃗ )
𝑒
⃗ =
𝑎
⃗
|𝑎
⃗|
Contoh :
1. Tentukan vektor satuan dari 𝑏
⃗⃗ =[
0
3
−4
]!
Jawab: 𝑒
⃗ =
𝑏
⃗⃗
|𝑏
⃗⃗|
==
[
0
3
−4
]
√02+32(−4)
2
=
[
0
3
−4
]
√0+9+16
=
[
0
3
−4
]
√25
=
[
0
3
−4
]
5
=
[
0
5
3
5
−
4
5]
=[
0
3
5
−
4
5
]
2. Tentukan vektor satuan dari 𝑐
⃗ =2𝑖
⃗+ 𝑗
⃗ + 𝑘
⃗⃗
Jawab: 𝑐
⃗ =2𝑖
⃗ + 𝑗
⃗ + 𝑘
⃗⃗=[
𝑖
𝑗
𝑘
] = [
2
1
1
], maka 𝑒
⃗ =
𝑐
⃗
|𝑐
⃗|
=
[
2
1
1
]
√22
+12
+12
=
[
2
1
1
]
√4+1+1
=
[
2
1
1
]
√6
=
[
2
√6
1
√6
1
√6]
B. Perkalian vektor dengan skalar
Contoh:
Diketahui 𝑎
⃗ = [
2
0
−4
] 𝑑𝑎𝑛 𝑏
⃗⃗ = [
1
8
−5
]
Tentukan ;
1. 3𝑎
⃗
2. −2𝑏
⃗⃗
Jawab:
1. 3𝑎
⃗ =3. [
2
0
−4
] = [
6
0
−12
]
2. −2𝑏
⃗⃗=(-2). [
1
8
−5
] = [
−2
−16
10
]
C. Penjumalahan Vektor
Contoh:
Diketahui 𝑎
⃗ = [
2
4
3
] 𝑑𝑎𝑛 𝑏
⃗⃗ = [
1
−5
−4
] , 𝑚𝑎𝑘𝑎:
1. 𝑎
⃗ + 𝑏
⃗⃗ =[
2
4
3
] + [
1
−5
−4
] = [
3
−1
−1
]
2. 3𝑎
⃗⃗ + 𝑏
⃗⃗ =3. [
2
4
3
] + [
1
−5
−4
] = [
6
12
9
] + [
1
−5
−4
] = [
7
7
5
]
D. Selisih/pengurangan Vektor
Contoh:
Diketahui 𝑎
⃗ = [
2
4
1
] 𝑑𝑎𝑛 𝑏
⃗⃗ = [
1
−5
3
] , 𝑚𝑎𝑘𝑎:
1. 𝑎
⃗ − 𝑏
⃗⃗ =[
2
4
1
] − [
1
−5
3
] = [
1
10
−2
]
2. 𝑎
⃗⃗ − 2𝑏
⃗⃗ =[
2
1
1
] − 2. [
1
−5
3
] = [
2
4
1
] − [
2
−10
6
] = [
0
14
−5
]

3. E. Materi Tambahan
Contoh;
1. Tentukan vektor satuannya
a. 𝑑
⃗ = 4𝑖
⃗− 𝑗
⃗ + 𝑘
⃗⃗
b. 𝑝
⃗ = −4𝑗
⃗− 2𝑘
⃗⃗
c. 𝑞
⃗ = 𝑖
⃗ − 5𝑘
⃗⃗
Jawab.
a. 𝑑
⃗ = 4𝑖
⃗− 𝑗
⃗ + 𝑘
⃗⃗ = [
𝑖
𝑗
𝑘
] = [
4
−1
1
], maka 𝑒
⃗ =
𝑐
⃗
|𝑑
⃗|
=
[
4
−1
1
]
√42+(−1)2
+12
=
[
4
−1
1
]
√16+1+1
=
[
4
−1
1
]
√18
=
[
4
√18
−
1
√18
1
√18 ]
b. 𝑝
⃗ = −4𝑗
⃗− 2𝑘
⃗⃗=[
𝑖
𝑗
𝑘
] = [
0
−4
−2
], maka 𝑒
⃗ =
𝑐
⃗
|𝑝
⃗|
=
[
0
−4
−2
]
√02
+(−4)2
+(−2)2
=
[
0
−4
−2
]
√0+16+4
=
[
0
−4
−2
]
√20
=
[
0
√20
−
4
√20
−
2
√20]
= [
0
−
4
√20
−
2
√20
]
c. 𝑞
⃗ = 𝑖
⃗ − 5𝑘
⃗⃗ = [
𝑖
𝑗
𝑘
] = [
1
0
−5
], maka 𝑒
⃗ =
𝑐
⃗
|𝑞
⃗⃗|
=
[
1
0
−5
]
√12
+02
+(−5)2
=
[
1
0
−5
]
√1+0+25
=
[
1
0
−5
]
√26
=
[
1
√26
0
√26
−
5
√26]
= [
1
√26
0
−
5
√26
]
 Latihan Soal
1. Tentukan vektor satuannya
a. 𝑟
⃗ =[
−6
8
]
b. 𝑏
⃗⃗ = [
0
4
]
c. 𝑐
⃗ = 3𝑖
⃗ + 𝑗
⃗ + 𝑘
⃗⃗
d. 𝑑
⃗ = 𝑖
⃗ − 𝑗
⃗ + 𝑘
⃗⃗
e. 𝑝
⃗ = −4𝑗
⃗ + 2𝑘
⃗⃗
---Selamat mengerjakan---