BAB IV VEKTOR

1. Hitunglah panjang dari vektor-vektor berikut

a. 𝑘 = [6

−8]

b. 𝑙 = [911

]

c. 𝑚 = [46

−2]

d. 𝑛 = [69

−7]

Jawaban :

𝑘 = [6

−8]

|𝑘| = √62 + 82

= √36 + 64 = √100 = 10

𝑙 = [911

]

|𝑙| = √92 + 112

√81 + 121 = √202

𝑚 = [46

−2]

|𝑚| = √42 + 62 + −22

√16 + 36 + 4 = √56 = 2√14

𝑛 = [69

−7]

|𝑛| = √62 + 92 + −72

√36 + 81 + 49 = √166

2. Hitunglah vektor satuan dari vektor-vektor berikut:

a. 𝑎 = [48]

b. 𝑏 = [−57

]

c. 𝑐 = [83

−1]

BAB IV VEKTOR

d. 𝑑 = [53

−8]

Jawaban:

a) |𝑎| = √42 + 82 = 4√5

𝑒 =𝑎

|𝑎|=

1

4√5[48]

=

[ 1

√52

√5]

b) |𝑏| = √−52 + 72 = √74

𝑒 =𝑏

|𝑏|=

1

√74[−57

]

=

[ −5

√747

√74]

c) |𝑐| = √82 + 32 + −12 = √74

𝑒 =𝑐

|𝑐|=

1

√74[

83

−1]

=

[

8

√743

√74−1

√74]

d) |𝑑| = √52 + 32 + −82 = √25 + 9 + 64 = 7√2

𝑒 =𝑑

|𝑑|=

1

7√2[

53

−8]

BAB IV VEKTOR

=

[

5

7√23

7√2−8

7√2]

3. Diketahui 𝑎 = 6𝑖 − 𝑥𝑗 + 10𝑘 dan 𝑏 = 8𝑖 + 7𝑗 − 9𝑘. Tentukanlah nilai x agar

vektor 𝑎 tegak lurus dengan vektor 𝑏 .

Jawaban:

𝑎 = 6𝑖 − 𝑥𝑗 + 10𝑘

𝑏 = 8𝑖 + 7𝑗 − 9𝑘

vektor 𝑎tegak lurus dengan vektor 𝑏,maka:

𝑎 . 𝑏 = 0

(6𝑖 − 𝑥𝑗 + 10𝑘)(8𝑖 + 7𝑗 − 9𝑘) = 0

48 − 7𝑥 − 90 = 0

−7𝑥 = 42

𝑥 = −6

4. Diketahui |𝑎| = 8 , |𝑏| = 9, sudut yang dibentuk kedua vektor 30°.

Tentukanlah 𝑎 . 𝑏

Jawaban:

𝑎 . 𝑏 = |𝑎|.|𝑏|cos30°

= 8 . 9 cos 30°

= 721

2√3 = 36√3

BAB IV VEKTOR

5. Diketahui 𝑎 = [−345

] 𝑑𝑎𝑛 𝑏 = [7

−26

] . Tentukanlah 𝑎 . 𝑏

Jawaban:

𝑎 . 𝑏 = [−345

] [7

−26

]

= (−21) + (−8) + 30 = 1

6. Jika 𝑎 = 6𝑖 + 7𝑗 − 2𝑘 𝑑𝑎𝑛 𝑏 = 4𝑖 + 7𝑗 − 9𝑘. Tentukanlah 𝑎 x 𝑏

Jawaban:

𝑎 x 𝑏 = |𝑖 𝑗 𝑘6 7 −24 7 −9

|

= (−63𝑖 + (−8)𝑗 + 42𝑘) − (−141𝑖 + (−54)𝑗 + 28𝑘)

= −49𝑖 + 46𝑗 + 14𝑘

7. Hitunglah besar sudut antara 𝑎 = 3𝑖 − 4𝑗 + 6𝑘 𝑑𝑎𝑛 𝑏 = 5𝑖 + 2𝑗 − 3𝑘

Jawaban:

𝑐𝑜𝑠𝜃 =𝑎 . 𝑏

|𝑎|. |𝑏|

=3(5) + (−4)2 + 6(−3)

√32 + −42 + 62. √52 + 22 + −32

= −11

48,15= −0,23

𝑐𝑜𝑠𝜃 = −0,23

𝜃 = 𝑎𝑟𝑐 cos(−0,23)

𝜃 = 103,3°

8. Diketahui 𝑎 = 5𝑖 − 𝑗 + 2𝑘 𝑑𝑎𝑛 𝑏 = 3𝑖 + 4𝑗 − 5𝑘.

BAB IV VEKTOR

Tentukanlah:

a) Proyeksi vektor 𝑎 pada vektor 𝑏

b) Panjang proyeksi vektor 𝑎 pada vektor 𝑏

Jawaban:

a) Proyeksi vektor 𝑎pada vektor 𝑏

(𝑎 . 𝑏)

|𝑏|𝟐 𝑏 =

5(3) + (−1)4 + 2(−5)

(√32 + 42 + −52)2 (3𝑖 + 4𝑗 − 5𝑘)

=1(3𝑖 + 4𝑗 − 5𝑘)

(√50)2

=3

50𝑖 +

2

25𝑗 −

1

10𝑘

b) Panjang proyeksi vektor 𝑎pada vektor𝑏

𝑎 . 𝑏

|𝑏|=

1

√50=

1

5√2

1

5√2 .5√2

5√2=

5√2

50=

√2

10