Roboguru

Diketahui vektor u=⎝⎛​25−8​⎠⎞​ dan v=⎝⎛​0−34​⎠⎞​. Jika w=3u+5v, hasil u−3v−2w adalah ...

Pertanyaan

Diketahui vektor begin mathsize 14px style straight u with rightwards arrow on top equals open parentheses table row 2 row 5 row cell negative 8 end cell end table close parentheses end style dan begin mathsize 14px style straight v with rightwards arrow on top equals open parentheses table row 0 row cell negative 3 end cell row 4 end table close parentheses end style. Jika begin mathsize 14px style straight w with rightwards arrow on top equals 3 straight u with rightwards arrow on top plus 5 straight v with rightwards arrow on top end style, hasil begin mathsize 14px style straight u with rightwards arrow on top minus 3 straight v with rightwards arrow on top minus 2 straight w with rightwards arrow on top end style adalah ...

Pembahasan Soal:

Dengan menggunakan konsep perkalian skalar dengan vektor dan penjumlahanan pada vektor dalam ruang dimensi tiga maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight w with rightwards arrow on top end cell equals cell 3 straight u with rightwards arrow on top plus 5 straight v with rightwards arrow on top end cell row blank equals cell 3 open parentheses table row 2 row 5 row cell negative 8 end cell end table close parentheses plus 5 open parentheses table row 0 row cell negative 3 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses table row 6 row 15 row cell negative 24 end cell end table close parentheses plus open parentheses table row 0 row cell negative 15 end cell row 20 end table close parentheses end cell row blank equals cell open parentheses table row cell 6 plus 0 end cell row cell 15 plus open parentheses negative 15 close parentheses end cell row cell negative 24 plus 20 end cell end table close parentheses end cell row blank equals cell open parentheses table row 6 row 0 row cell negative 4 end cell end table close parentheses end cell end table end style 

maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight u with rightwards arrow on top minus 3 straight v with rightwards arrow on top minus 2 straight w with rightwards arrow on top end cell equals cell open parentheses table row 2 row 5 row cell negative 8 end cell end table close parentheses minus 3 open parentheses table row 0 row cell negative 3 end cell row 4 end table close parentheses minus 2 open parentheses table row 6 row 0 row cell negative 4 end cell end table close parentheses end cell row blank equals cell open parentheses table row 2 row 5 row cell negative 8 end cell end table close parentheses minus open parentheses table row 0 row cell negative 9 end cell row 12 end table close parentheses minus open parentheses table row 12 row 0 row cell negative 8 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 2 minus 0 minus 12 end cell row cell 5 minus open parentheses negative 9 close parentheses minus 0 end cell row cell negative 8 minus 12 minus open parentheses negative 8 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 10 end cell row 14 row cell negative 12 end cell end table close parentheses end cell end table end style 

Dengan demikian, hasil dari begin mathsize 14px style straight u with rightwards arrow on top minus 3 straight v with rightwards arrow on top minus 2 straight w with rightwards arrow on top end style adalah begin mathsize 14px style open parentheses table row cell negative 10 end cell row 14 row cell negative 12 end cell end table close parentheses end style

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Mariyam

Mahasiswa/Alumni Institut Pertanian Bogor

Terakhir diupdate 12 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Jika a=2i+j​−4kdanb=3i−2j​−5k, hasil operasi 5a−3b adalah ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 5 straight a with rightwards arrow on top minus 3 straight b with rightwards arrow on top end cell equals cell 5 left parenthesis 2 straight i with rightwards arrow on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis minus 3 left parenthesis 3 straight i with rightwards arrow on top minus 2 straight j with hat on top minus 5 straight k with hat on top right parenthesis end cell row blank equals cell straight i with rightwards arrow on top plus 11 straight j with hat on top minus 5 straight k with hat on top end cell end table end style

Jadi, jawaban yang tepat adalah B.

0

Roboguru

Diketahui vektor a=4i−23​j​+2k; b=2i+5j​−3k dan c=2j​−3k. Hasil dari a−21​b+21​c 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 row cell straight a subscript 3 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 plus straight a subscript 3 straight k 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 row cell straight b subscript 3 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 plus straight b subscript 3 straight k 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 straight a subscript 3 plus-or-minus straight b subscript 3 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 plus open parentheses straight a subscript 3 plus-or-minus straight b subscript 3 close parentheses straight k 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 row cell za subscript 3 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 plus za subscript 3 straight k with rightwards arrow on top

Berdasarkan sifat-sifat operasi vektor tersebut, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight a with rightwards arrow on top minus 1 half straight b with rightwards arrow on top plus 1 half straight c with rightwards arrow on top end cell equals cell open parentheses table row 4 row cell negative begin inline style 3 over 2 end style end cell row 2 end table close parentheses minus 1 half open parentheses table row 2 row 5 row cell negative 3 end cell end table close parentheses plus 1 half open parentheses table row 0 row 2 row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 4 row cell negative begin inline style 3 over 2 end style end cell row 2 end table close parentheses minus open parentheses table row 1 row cell begin inline style 5 over 2 end style end cell row cell negative begin inline style 3 over 2 end style end cell end table close parentheses plus open parentheses table row 0 row 1 row cell negative begin inline style 3 over 2 end style end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 4 minus 1 end cell row cell negative begin inline style 3 over 2 end style minus begin inline style 5 over 2 end style plus 1 end cell row cell 2 plus begin inline style 3 over 2 end style minus begin inline style 3 over 2 end style end cell end table close parentheses end cell row blank equals cell open parentheses table row 3 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell 3 straight i with rightwards arrow on top minus 3 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top end cell end table 

Dengan demikian, hasil operasi vektor pada soal adalah 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 i 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 j 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 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight k with rightwards arrow on top end cell end table.

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Diketahui vektor a=3i+21​j​−21​k, b=i+5j​−k, dan  c=23​i. Hasil dari a+21​b−c adalah ...

Pembahasan Soal:

Diketahui bahwa vektor a with rightwards arrow on top,b with rightwards arrow on top, dan c with rightwards arrow on top merupakan vektor basis (i,j,k) yang berada pada dimensi tiga yang dapat diubah ke vektor baris berada pada dimensi tiga sehingga diperoleh

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

maka

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top plus 1 half b with rightwards arrow on top minus c with rightwards arrow on top end cell equals cell open parentheses 3 comma 1 half comma negative 1 half close parentheses plus 1 half open parentheses 1 , 5 comma negative 1 close parentheses minus open parentheses 3 over 2 comma 0 , 0 close parentheses end cell row blank equals cell open parentheses 3 comma 1 half comma negative 1 half close parentheses plus open parentheses 1 half comma 5 over 2 comma negative 1 half close parentheses minus open parentheses 3 over 2 comma 0 , 0 close parentheses end cell row blank equals cell open parentheses 3 1 half comma 6 over 2 comma negative 2 over 2 close parentheses minus open parentheses 3 over 2 comma 0 , 0 close parentheses end cell row blank equals cell open parentheses 7 over 2 comma 6 over 2 comma negative 2 over 2 close parentheses minus open parentheses 3 over 2 comma 0 , 0 close parentheses end cell row blank equals cell open parentheses 4 over 2 comma 6 over 2 comma negative 2 over 2 close parentheses end cell row blank equals cell open parentheses 2 , 3 comma negative 1 close parentheses end cell end table

Jadi, hasil dari a with rightwards arrow on top plus 1 half b with rightwards arrow on top minus c with rightwards arrow on top adalah open parentheses 2 , 3 comma negative 1 close parentheses.

1

Roboguru

Diketahui vektor a=3i+21​j​−21​k; b=i+5j​−k dan c=23​i. Hasil dari a+21​b−c 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 row cell straight a subscript 3 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 plus straight a subscript 3 straight k 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 row cell straight b subscript 3 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 plus straight b subscript 3 straight k 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 straight a subscript 3 plus-or-minus straight b subscript 3 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 plus open parentheses straight a subscript 3 plus-or-minus straight b subscript 3 close parentheses straight k 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 row cell za subscript 3 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 plus za subscript 3 straight k with rightwards arrow on top

Berdasarkan sifat-sifat operasi vektor tersebut, maka:

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

Dengan demikian, hasil operasi vektor pada soal adalah 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 i 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 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight j 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 cell straight k with rightwards arrow on top end cell end table.

Oleh karena itu, jawaban yang benar adalah E.

0

Roboguru

Diketahui vektor a=⎝⎛​3−21​⎠⎞​,b=⎝⎛​−345​⎠⎞​ dan c=⎝⎛​5−32​⎠⎞​. Tentukanlah: c. 5a−3c

Pembahasan Soal:

Diketahui vektor begin mathsize 14px style a with rightwards arrow on top equals open parentheses table row 3 row cell negative 2 end cell row 1 end table close parentheses comma space b with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 4 row 5 end table close parentheses end style dan begin mathsize 14px style c with rightwards arrow on top equals open parentheses table row 5 row cell negative 3 end cell row 2 end table close parentheses end style, maka:

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 5 a with rightwards arrow on top minus 3 c with rightwards arrow on top end cell equals cell 5 times open parentheses table row 3 row cell negative 2 end cell row 1 end table close parentheses minus 3 times open parentheses table row 5 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell open parentheses table row cell 5 times 3 end cell row cell 5 times left parenthesis negative 2 right parenthesis end cell row cell 5 times 1 end cell end table close parentheses minus open parentheses table row cell 3 times 5 end cell row cell 3 times left parenthesis negative 3 right parenthesis end cell row cell 3 times 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row 15 row cell negative 10 end cell row 5 end table close parentheses minus open parentheses table row 15 row cell negative 9 end cell row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell 15 minus 15 end cell row cell negative 10 minus left parenthesis negative 9 right parenthesis end cell row cell 5 minus 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row 0 row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell end table end style 

Jadi, hasil dari begin mathsize 14px style 5 a with rightwards arrow on top minus 3 c 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 cell open parentheses table row 0 row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell end table end style.

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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