Roboguru

Diketahui  dan . Tentukanlah: .

Pertanyaan

Diketahui a with rightwards arrow on top equals 3 i with rightwards arrow on top minus 2 j with rightwards arrow on top plus k with rightwards arrow on top dan b with rightwards arrow on top equals i with rightwards arrow on top plus 3 j with rightwards arrow on top minus 2 k with rightwards arrow on top.

Tentukanlah: a with rightwards arrow on top minus b with rightwards arrow on top.space 

 

Pembahasan Soal:

Rumus penjumlahan vektor a with rightwards arrow on top equals x subscript 1 i with rightwards arrow on top plus x subscript 2 j with rightwards arrow on top plus x subscript 3 k with rightwards arrow on top dan b with rightwards arrow on top equals y subscript 1 i with rightwards arrow on top plus y subscript 2 j with rightwards arrow on top plus y subscript 3 k with rightwards arrow on top adalah 

a with rightwards arrow on top minus b with rightwards arrow on top equals open parentheses x subscript 1 minus y subscript 1 close parentheses i with rightwards arrow on top plus open parentheses x subscript 2 minus y subscript 2 close parentheses j with rightwards arrow on top plus open parentheses x subscript 3 minus y subscript 3 close parentheses k with rightwards arrow on top

Dengan menyelesaikan permasalahan pada soal,

a with rightwards arrow on top equals 3 i with rightwards arrow on top minus 2 j with rightwards arrow on top plus k with rightwards arrow on top equals 3 i with rightwards arrow on top plus open parentheses negative 2 close parentheses j with rightwards arrow on top plus k with rightwards arrow on top
b with rightwards arrow on top equals i with rightwards arrow on top plus 3 j with rightwards arrow on top minus 2 k with rightwards arrow on top equals i with rightwards arrow on top plus 3 j with rightwards arrow on top plus open parentheses negative 2 close parentheses k with rightwards arrow on top 

menggunakan rumus di atas, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top minus b with rightwards arrow on top end cell equals cell open parentheses 3 i with rightwards arrow on top plus open parentheses negative 2 close parentheses j with rightwards arrow on top plus k with rightwards arrow on top close parentheses plus open parentheses i with rightwards arrow on top plus 3 j with rightwards arrow on top plus open parentheses negative 2 close parentheses k with rightwards arrow on top close parentheses end cell row blank equals cell open parentheses 3 minus 1 close parentheses i with rightwards arrow on top plus open parentheses negative 2 minus 3 close parentheses j with rightwards arrow on top plus open parentheses 1 minus open parentheses negative 2 close parentheses close parentheses k with rightwards arrow on top end cell row blank equals cell open parentheses 2 close parentheses i with rightwards arrow on top plus open parentheses negative 5 close parentheses j with rightwards arrow on top plus open parentheses 3 close parentheses k with rightwards arrow on top end cell row blank equals cell 2 i with rightwards arrow on top minus 5 j with rightwards arrow on top plus 3 k with rightwards arrow on top end cell end table

Jadi, table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top minus b with rightwards arrow on top end cell equals cell 2 i with rightwards arrow on top minus 5 j with rightwards arrow on top plus 3 k with rightwards arrow on top end cell end table.space 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Ridha

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 05 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui titik-titik . Jika vector  dan , maka vektor  adalah ...

Pembahasan Soal:

Dari soal diketahui

straight A left parenthesis 2 comma 5 comma 2 right parenthesis comma space straight B left parenthesis 3 comma 2 comma negative 1 right parenthesis comma space straight C left parenthesis 2 comma 2 comma 2 right parenthesis  

Maka vektor space straight u equals AB adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell straight u with rightwards arrow on top end cell equals AB row blank equals cell b minus a end cell row blank equals cell open parentheses 3 comma 2 comma negative 1 close parentheses minus open parentheses 2 comma 5 comma 2 close parentheses end cell row blank equals cell open parentheses 1 comma negative 3 comma negative 3 close parentheses end cell end table 

vektor space straight v equals CA  adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell vektor space straight v end cell equals CA row blank equals cell a minus c end cell row blank equals cell open parentheses 2 comma 5 comma 2 close parentheses minus open parentheses 2 comma 2 comma 2 close parentheses end cell row blank equals cell open parentheses 0 comma 3 comma 0 close parentheses end cell end table

Dengan demikian w with rightwards arrow on top equals v with rightwards arrow on top minus u with rightwards arrow on top

table attributes columnalign right center left columnspacing 0px end attributes row cell w with rightwards arrow on top end cell equals cell v with rightwards arrow on top minus u with rightwards arrow on top end cell row blank equals cell open parentheses 1 comma negative 3 comma negative 3 close parentheses minus open parentheses 0 comma 3 comma 0 close parentheses end cell row blank equals cell open parentheses 1 comma negative 6 comma negative 3 close parentheses end cell end table 

Jadi, w with rightwards arrow on top adalah i minus 6 j minus 3 k 

0

Roboguru

Jika , dan , maka panjang vektor adalah ...

Pembahasan Soal:

Misalkan terdapat vektor straight a with rightwards arrow on top equals open parentheses table row cell straight a subscript 1 end cell row cell straight a subscript 2 end cell end table close parentheses equals straight a subscript 1 straight i with rightwards arrow on top plus straight a subscript 2 straight j with rightwards arrow on top, straight b with rightwards arrow on top equals open parentheses table row cell straight b subscript 1 end cell row cell straight b subscript 2 end cell end table close parentheses equals straight b subscript 1 straight i with rightwards arrow on top plus straight b subscript 2 straight j with rightwards arrow on top dan straight z adalah skalar. maka berlaku:

open parentheses straight i close parentheses space straight a with rightwards arrow on top plus-or-minus straight b with rightwards arrow on top equals open parentheses table row cell straight a subscript 1 plus-or-minus straight b subscript 1 end cell row cell straight a subscript 2 plus-or-minus straight b subscript 2 end cell end table close parentheses equals open parentheses straight a subscript 1 plus-or-minus straight b subscript 1 close parentheses straight i with rightwards arrow on top plus open parentheses straight a subscript 2 plus-or-minus straight b subscript 2 close parentheses straight j with rightwards arrow on top left parenthesis ii right parenthesis space straight z straight a with rightwards arrow on top equals open parentheses table row cell za subscript 1 end cell row cell za subscript 2 end cell end table close parentheses equals za subscript 1 straight i with rightwards arrow on top plus za subscript 2 straight j with rightwards arrow on top

Selain itu, untuk mencari panjang vektor straight a with rightwards arrow on top equals open parentheses table row cell straight a subscript 1 end cell row cell straight a subscript 2 end cell end table close parentheses equals straight a subscript 1 straight i with rightwards arrow on top plus straight a subscript 2 straight j with rightwards arrow on top adalah:

open vertical bar straight a with rightwards arrow on top close vertical bar equals square root of straight a subscript 1 squared plus straight a subscript 2 squared end root .

Berdasarkan uraian tentang vektor tersebut, maka:

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

Panjang vektor table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight d 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 cell open parentheses table row 5 row 12 end table close parentheses end cell end table adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight d with rightwards arrow on top close vertical bar end cell equals cell square root of 5 squared plus 12 squared end root end cell row blank equals cell square root of 25 plus 144 end root end cell row blank equals cell square root of 169 end cell row blank equals 13 end table

Dengan demikian, panjang vektor straight d with rightwards arrow on top equals 3 straight a with rightwards arrow on top minus 2 straight b with rightwards arrow on top plus straight c with rightwards arrow on top yaitu 13.

Oleh karena itu, jawaban yang benar adalah C.

0

Roboguru

Diketahui vektor , vektor , dan vektor . Jika , dengan  bilangan real, maka nilai  adalah ...

Pembahasan Soal:

Nilai k pada operasi hitung vektor di atas adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 a with rightwards arrow on top minus 3 b with rightwards arrow on top plus k c with rightwards arrow on top end cell equals cell open parentheses table row 0 row cell negative 4 end cell end table close parentheses end cell row cell 2 open parentheses table row 3 row cell negative 2 end cell end table close parentheses minus 3 open parentheses table row 4 row 1 end table close parentheses plus k open parentheses table row cell negative 2 end cell row cell negative 1 end cell end table close parentheses end cell equals cell open parentheses table row 0 row cell negative 4 end cell end table close parentheses end cell row cell open parentheses table row 6 row cell negative 4 end cell end table close parentheses minus open parentheses table row 12 row 3 end table close parentheses plus open parentheses table row cell negative 2 k end cell row cell negative k end cell end table close parentheses end cell equals cell open parentheses table row 0 row cell negative 4 end cell end table close parentheses end cell row cell open parentheses table row cell negative 6 minus 2 k end cell row cell negative 7 minus k end cell end table close parentheses end cell equals cell open parentheses table row 0 row cell negative 4 end cell end table close parentheses end cell end table

Dari kesamaan vektor di atas diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 6 minus 2 k end cell equals 0 row cell negative 2 k end cell equals 6 row k equals cell negative 3 end cell end table

Oleh karena itu, jawaban yang tepat adalah B.

0

Roboguru

Diketahui vektor , , dan . Hasil dari  adalah ....

Pembahasan Soal:

Dengan menerapkan konsep penjumlahan dan pengurangan vektor serta perkalian vektor dengan skalar, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 u with rightwards arrow on top plus v with rightwards arrow on top minus w with rightwards arrow on top end cell equals cell 2 open parentheses table row 3 row cell negative 1 end cell row 2 end table close parentheses plus open parentheses table row 1 row 4 row cell negative 3 end cell end table close parentheses minus open parentheses table row 5 row 2 row 1 end table close parentheses end cell row blank equals cell open parentheses table row 6 row cell negative 2 end cell row 4 end table close parentheses plus open parentheses table row 1 row 4 row cell negative 3 end cell end table close parentheses minus open parentheses table row 5 row 2 row 1 end table close parentheses end cell row blank equals cell open parentheses table row 2 row 0 row 0 end table close parentheses end cell row blank equals cell 2 i with hat on top end cell end table end style 

Jadi, hasil dari begin mathsize 14px style 2 u with rightwards arrow on top plus v with rightwards arrow on top minus w with rightwards arrow on top end style adalah begin mathsize 14px style 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 i with hat on top end cell end table end style.

Dengan demikian, jawaban yang tepat adalah A.

0

Roboguru

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

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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