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Diketahui vektor  dan , vektor . Tentukan: a) vektor  b) vektor

Pertanyaan

Diketahui vektor a with rightwards arrow on top equals 5 i with hat on top minus 12 j with hat on top dan b with rightwards arrow on top equals left parenthesis 3 comma space 4 right parenthesis, vektor c with rightwards arrow on top equals negative 2 i with hat on top minus 6 j with hat on top. Tentukan:

a) vektor a with rightwards arrow on top minus 2 b with rightwards arrow on top minus c with rightwards arrow on top

b) vektor 2 a with rightwards arrow on top plus b with rightwards arrow on top minus 1 half c with rightwards arrow on top

Pembahasan Soal:

Ingat kemabli konsep operasi hitung vektor.

Untuk a dan b vektor-vektor pada dua dimensi, berlaku:

a±b==(a1,a2)+(b1,b2)(a1+b1,a2+b2)

Perkalian skalar dengan vektor.

Jika k skalar tak nol dan vektor a=(a1,a2,...an), maka ka=(ka1,ka2,...kan).

Diketahui:

vektor a with rightwards arrow on top equals 5 i with hat on top minus 12 j with hat on top dan b with rightwards arrow on top equals left parenthesis 3 comma space 4 right parenthesis equals 3 i with hat on top plus 4 j with hat on top, vektor c with rightwards arrow on top equals negative 2 i with hat on top minus 6 j with hat on top.

a) menentukan vektor a with rightwards arrow on top minus 2 b with rightwards arrow on top minus c with rightwards arrow on top

table attributes columnalign right center left columnspacing 2px end attributes row cell a with rightwards arrow on top minus 2 b with rightwards arrow on top minus c with rightwards arrow on top end cell equals cell open parentheses 5 i with hat on top minus 12 j with hat on top close parentheses minus 2 open parentheses 3 i with hat on top minus 4 j with hat on top close parentheses minus open parentheses negative 2 i with hat on top minus 6 j with hat on top close parentheses end cell row blank equals cell 5 i with hat on top minus 12 j with hat on top minus 6 i with hat on top plus 8 j with hat on top plus 2 i with hat on top plus 6 j with hat on top end cell row cell a with rightwards arrow on top minus 2 b with rightwards arrow on top minus c with rightwards arrow on top end cell equals cell i with hat on top plus 2 j with hat on top end cell end table

b) menentukan vektor 2 a with rightwards arrow on top plus b with rightwards arrow on top minus 1 half c with rightwards arrow on top

table attributes columnalign right center left columnspacing 2px end attributes row cell 2 a with rightwards arrow on top plus b with rightwards arrow on top minus 1 half c with rightwards arrow on top end cell equals cell 2 open parentheses 5 i with hat on top minus 12 j with hat on top close parentheses plus open parentheses 3 i with hat on top minus 4 j with hat on top close parentheses minus 1 half open parentheses negative 2 i with hat on top minus 6 j with hat on top close parentheses end cell row blank equals cell 10 i with hat on top minus 24 j with hat on top plus 3 i with hat on top minus 4 j with hat on top plus i with hat on top plus 3 j with hat on top end cell row cell 2 a with rightwards arrow on top plus b with rightwards arrow on top minus 1 half c with rightwards arrow on top end cell equals cell 14 i with hat on top minus 25 j with hat on top end cell end table

Jadi, diperoleh a with rightwards arrow on top minus 2 b with rightwards arrow on top minus c with rightwards arrow on top equals i with hat on top plus 2 j with hat on top dan 2 a with rightwards arrow on top plus b with rightwards arrow on top minus 1 half c with rightwards arrow on top equals 14 i with hat on top minus 25 j with hat on top.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Difhayanti

Mahasiswa/Alumni Universitas Muhammadiyah Prof. Dr. Hamka

Terakhir diupdate 25 September 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui vektor , vektor , dan vektor . Jika  nilai  yang memenuhi adalah ....

Pembahasan Soal:

Gunakan konsep kesamaan dua vektor, penjumlahan, pengurangan dan perkalian vektor.

Agar soal menjadi benar, asumsikan bahwa vektor q with rightwards arrow on top equals open parentheses table row 5 row 4 end table close parentheses seharusnya adalah vektor r with rightwards arrow on top equals open parentheses table row 5 row 4 end table close parentheses.

Diketahui vektor p with rightwards arrow on top equals open parentheses table row 6 row cell negative 2 end cell end table close parentheses, vektor q with rightwards arrow on top equals open parentheses table row cell negative 1 end cell row cell negative 3 end cell end table close parentheses, dan vektor r with rightwards arrow on top equals open parentheses table row 5 row 4 end table close parentheses. x p with rightwards arrow on top minus 3 q with rightwards arrow on top plus 3 r with rightwards arrow on top equals open parentheses table row 21 row 20 end table close parentheses. Akan ditentukan nilai x.

Perhatikan perhitungan berikut.

table attributes columnalign right center left columnspacing 2px end attributes row cell x p with rightwards arrow on top minus 3 q with rightwards arrow on top plus 3 r with rightwards arrow on top end cell equals cell open parentheses table row 21 row 20 end table close parentheses end cell row cell x open parentheses table row 6 row cell negative 2 end cell end table close parentheses minus 3 open parentheses table row cell negative 1 end cell row cell negative 3 end cell end table close parentheses plus 3 open parentheses table row 5 row 4 end table close parentheses end cell equals cell open parentheses table row 21 row 20 end table close parentheses end cell row cell open parentheses table row cell 6 x end cell row cell negative 2 x end cell end table close parentheses plus open parentheses table row 3 row 9 end table close parentheses plus open parentheses table row 15 row 12 end table close parentheses end cell equals cell open parentheses table row 21 row 20 end table close parentheses end cell row cell open parentheses table row cell 6 x plus 3 plus 15 end cell row cell negative 2 x plus 9 plus 12 end cell end table close parentheses end cell equals cell open parentheses table row 21 row 20 end table close parentheses end cell row cell open parentheses table row cell 6 x plus 18 end cell row cell negative 2 x plus 21 end cell end table close parentheses end cell equals cell open parentheses table row 21 row 20 end table close parentheses end cell row cell 6 x plus 18 end cell equals 21 row cell 6 x end cell equals cell 21 minus 18 end cell row cell 6 x end cell equals 3 row x equals cell 3 over 6 end cell row x equals cell 1 half end cell end table

Diperoleh nilai x equals 1 half.

Jadi, jawaban yang tepat adalah C.

0

Roboguru

Diketahui . Jika , hasil  adalah ...

Pembahasan Soal:

Operasi perkalian vektor dengan skalar dan pengurangan dua vektor.

begin mathsize 14px style straight c with rightwards arrow on top equals 2 straight a with rightwards arrow on top minus 3 straight b with rightwards arrow on top straight c with rightwards arrow on top equals 2 open parentheses table row 3 row cell negative 2 end cell row 5 end table close parentheses minus 3 open parentheses table row 1 row 0 row cell negative 4 end cell end table close parentheses straight c with rightwards arrow on top equals open parentheses table row 6 row cell negative 4 end cell row 10 end table close parentheses minus open parentheses table row 3 row 0 row cell negative 12 end cell end table close parentheses straight c with rightwards arrow on top equals open parentheses table row 3 row cell negative 4 end cell row 22 end table close parentheses end style

Sehingga,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight a with rightwards arrow on top plus 3 straight b with rightwards arrow on top minus 2 straight c with rightwards arrow on top end cell equals cell open parentheses table row 3 row cell negative 2 end cell row 5 end table close parentheses plus 3 open parentheses table row 1 row 0 row cell negative 4 end cell end table close parentheses minus 2 open parentheses table row 3 row cell negative 4 end cell row 22 end table close parentheses end cell row blank equals cell open parentheses table row 3 row cell negative 2 end cell row 5 end table close parentheses plus open parentheses table row 3 row 0 row cell negative 12 end cell end table close parentheses minus open parentheses table row 6 row cell negative 8 end cell row 44 end table close parentheses end cell row blank equals cell open parentheses table row cell 3 plus 3 minus 6 end cell row cell negative 2 plus 0 plus 8 end cell row cell 5 minus 12 minus 44 end cell end table close parentheses end cell row blank equals cell open parentheses table row 0 row 8 row cell negative 51 end cell end table close parentheses end cell end table end style

Jadi, pilihan jawaban yang tepat adalah B.

0

Roboguru

Diketahui koordinat titik . Jika  berturut-turut vektor posisi titik A, B, dan C, hasil  adalah ...

Pembahasan Soal:

Karena  straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space dan space straight c with rightwards arrow on top berturut-turut vektor posisi titik A, B, dan C, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 straight a with rightwards arrow on top minus 3 straight b with rightwards arrow on top plus straight c end cell equals cell 2 open parentheses table row 4 row cell negative 3 end cell end table close parentheses minus 3 open parentheses table row cell negative 1 end cell row cell negative 5 end cell end table close parentheses plus open parentheses table row cell negative 2 end cell row 3 end table close parentheses end cell row blank equals cell open parentheses table row 8 row cell negative 6 end cell end table close parentheses minus open parentheses table row cell negative 3 end cell row cell negative 15 end cell end table close parentheses plus open parentheses table row cell negative 2 end cell row 3 end table close parentheses end cell row blank equals cell open parentheses table row 9 row 12 end table close parentheses end cell end table

Sehingga, table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight a with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight b with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight c end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row 9 row 12 end table close parentheses end cell end table

Jadi, pilihan jawaban yang tepat adalah A.

0

Roboguru

Vektor-vektor dari posisi A, B, dan C adalah , , . Hitunglah vektor

Pembahasan Soal:

Jika diketahui vektor a with rightwards arrow on top equals open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell end table close parentheses dan b with rightwards arrow on top equals open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell end table close parentheses, maka berlaku

a with rightwards arrow on top plus b with rightwards arrow on top equals open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell end table close parentheses plus open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell end table close parentheses equals open parentheses table row cell a subscript 1 plus b subscript 1 end cell row cell a subscript 2 plus b subscript 2 end cell end table close parentheses

a with rightwards arrow on top minus b with rightwards arrow on top equals open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell end table close parentheses minus open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell end table close parentheses equals open parentheses table row cell a subscript 1 minus b subscript 1 end cell row cell a subscript 2 minus b subscript 2 end cell end table close parentheses

k times a with rightwards arrow on top equals k open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell end table close parentheses equals open parentheses table row cell k times a subscript 1 end cell row cell k times a subscript 2 end cell end table close parentheses

 

Hasil dari operasi hitung vektor tersebut adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 a with rightwards arrow on top plus b with rightwards arrow on top minus 3 c with rightwards arrow on top end cell equals cell 2 open parentheses table row 4 row 3 end table close parentheses plus open parentheses table row cell negative 2 end cell row 3 end table close parentheses minus 3 open parentheses table row 1 row 5 end table close parentheses end cell row blank equals cell open parentheses table row 8 row 6 end table close parentheses plus open parentheses table row cell negative 2 end cell row 3 end table close parentheses minus open parentheses table row 3 row 15 end table close parentheses end cell row blank equals cell open parentheses table row cell 8 plus open parentheses negative 2 close parentheses minus 3 end cell row cell 6 plus 3 minus 15 end cell end table close parentheses end cell row blank equals cell open parentheses table row 3 row cell negative 6 end cell end table close parentheses end cell end table

Dengan demikian, hasil dari table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell a with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell b with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell c with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank i end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 6 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank j end table 

0

Roboguru

Diketahui vektor , vektor , jika  maka hasil dari  adalah ....

Pembahasan Soal:

Diketahui vektor begin mathsize 14px style straight a with rightwards arrow on top equals open parentheses table row 2 row 4 end table close parentheses end style, vektor begin mathsize 14px style straight b with rightwards arrow on top equals open parentheses table row cell negative 5 end cell row cell negative 7 end cell end table close parentheses end style, dan begin mathsize 14px style straight c with rightwards arrow on top equals 2 straight a with rightwards arrow on top plus straight b with rightwards arrow on top end style. Maka:

begin mathsize 14px style straight c with rightwards arrow on top equals 2 straight a with rightwards arrow on top plus straight b with rightwards arrow on top straight c with rightwards arrow on top equals 2 open parentheses table row 2 row 4 end table close parentheses plus open parentheses table row cell negative 5 end cell row cell negative 7 end cell end table close parentheses straight c with rightwards arrow on top equals open parentheses table row 4 row 8 end table close parentheses plus open parentheses table row cell negative 5 end cell row cell negative 7 end cell end table close parentheses straight c with rightwards arrow on top equals open parentheses table row cell negative 1 end cell row 1 end table close parentheses end style 

Sehingga:

 begin mathsize 14px style straight a with rightwards arrow on top minus 3 straight b with rightwards arrow on top plus 2 straight c with rightwards arrow on top equals open parentheses table row 2 row 4 end table close parentheses minus 3 open parentheses table row cell negative 5 end cell row cell negative 7 end cell end table close parentheses plus 2 open parentheses table row cell negative 1 end cell row 1 end table close parentheses space space space space space space space space space space space space space space space space space space space equals open parentheses table row 2 row 4 end table close parentheses minus open parentheses table row cell negative 15 end cell row cell negative 21 end cell end table close parentheses plus open parentheses table row cell negative 2 end cell row 2 end table close parentheses space space space space space space space space space space space space space space space space space space space equals open parentheses table row 15 row 27 end table close parentheses end style 

Jadi, jawaban yang benar adalah D.

0

Roboguru

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