This document discusses vectors and their operations including addition, subtraction, multiplication by scalars, dot products, projections, and applications to geometry. Some key points: - Vectors are represented by arrows with magnitude and direction. They can be added and subtracted by their components. - Scalar multiplication scales the magnitude of a vector. - The dot product of two vectors gives the cosine of the angle between them. It is used to find projections and determine if vectors are orthogonal. - Vector addition and subtraction is used to find the coordinates of points given other points and ratios of distances. The document provides definitions and formulas for common vector operations and gives examples of their applications in geometry problems.

2. VEKTOR
Laura Sari – Politeknik Negeri Cilacap

4. A
B
AB = a
– a
–½ a
2a
–2a
/ a
3
2

5. a
b
Penjumlahan Dua Vektor
a b
a + b

6. a
b
Pengurangan Dua Vektor
a
–b
a b

7. P(a, b, c) OP = p =
(
a
b
c
)= a i + b j + c k
^ ^ ^
O
Q(2, 4, 3)
x
y
z
a
b
c
2
4
3
OQ = q =
(
2
4
3
)= 2 i + 4 j + 3 k
^ ^ ^

8. A(a, b, c)
O
B(x, y, z)
x
y
z
OA + AB = OB
AB =
(
x
y
z
) (
a
b
c
)
– =
(
x – a
y – b
z – c
)
= (x – a) i + (y – b) j + (z – c) k
^
^ ^
AB
AB = OB – OA = b – a

9. A(3, 2, 3)
O
B(1, 3, 3)
x
y
z
AB = (
1
3
3
) (
3
2
3
)
– = (
–2
1
0
)
= –2 i + j
^ ^
AB
AB = OB – OA = b – a

10. a
b
Penjumlahan Dua Vektor
a
b
a + b
a =
(
a
b
c
) b =
(
x
y
z
)
; a + b =
(
a
b
c
) (
x
y
z
)
+ = (
a + x
b + y
c + z
)

11. a
b
Pengurangan Dua Vektor
a
–b
a – b
a =
(
a
b
c
) b =
(
x
y
z
)
; a – b =
(
a
b
c
) (
x
y
z
)
– = (
a – x
b – y
c – z
)

12. a
Perkalian Vektor & Skalar
a =
(
a
b
c
) ka = k
(
a
b
c
) (
ka
kb
kc
)
=
ka

13. a
a =
(
a
b
c
)
ka
a2 + b2 + c2
|a| =
k a2 + b2 + c2
|ka| = k|a| =

14. a
b
θ
a
b
a + b
θ
|a + b| = |a| + |b| + 2|a||b|cos
2 2 2

15. a
b
θ
a
–b
θ
|a – b| = |a| + |b| – 2|a||b|cos
2 2 2
a – b

16. a
a =
(
a
b
c
)
a
|a|
a =
(
a
b
c
)
a2 + b2 + c2
a =

17. O
A B
P
m n
AP : PB = m : n
a
p
b
O
A P
B
m n
a p
b
na + mb
m + n
p =
AP : PB = (m + n) : –n
–na + (m + n)b
(m + n) – n
p =

18. a
b
θ
a =
(
a
b
c
) b =
(
x
y
z
)
;
a b = |a||b|cos
a b = a  x + b  y + c  z
a b = b  a a a = |a|2

19. a
b
a = k b
a b = 0
a
b
Tegak Lurus Sejajar 3 Titik Segaris
A
C
B
AB = k BC

20. a
b
θ
a =
(
a
b
c
) b =
(
x
y
z
)
;
cos =
a b
|a||b|
a2 + b2 + c2 x2 + y2 + z2
cos =
a  x + b  y + c  z

21. a
b
θ
c

22. a
b
θ
c
a b
|b|
|c| =
x2 + y2 + z2
a  x + b  y + c  z
|c| =
|c| = |a| cos
a =
(
a
b
c
) b =
(
x
y
z
)
;

23. a
b
θ
c
a =
(
a
b
c
) b =
(
x
y
z
)
;
c = |c| b
a b
|b|
c = b
2
x2 + y2 + z2
a  x + b  y + c  z
c = (
x
y
z
)

24. Kuis
1. Diketahui 𝑢 = −3, 1, 2 , 𝑣 = 4, 0, −8 , carilah
2. Dua vektor 𝑢 dan 𝑣 besarnya 40 dan 20 satuan. Jika
sudut antara kedua vektor itu adalah 60⁰ , maka besar
dari 𝑢 − 𝑣 adalah …
3. Tentukan hasil perkalian titik dan sudut antara dari dua
vektor berikut
𝑢 = 4i – j + k
𝑣 = i + j + 2k
4. Misalkan 𝑎 = 4i +2j dan 𝑏 = 2i + 2j + k. Hitunglah
panjang proyeksi vektor 𝑎 pada 𝑏.
5. Jika koordinat titik A(10, -2, 3) dan B(5, 8, -12). Titik P
terletak diantara A dan B sedemikian sehingga AP : PB =
3 : 2. Tentukan koordinat titik P
a. 𝑢 − 𝑣
b. 6𝑢 + 2𝑣
c. 5(𝑣 − 4𝑢)

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