Roboguru

Jika a=(3−4​) dan b=(4−3​) maka ∣∣​b∣∣​a+∣a∣b adalah ....

Pertanyaan

Jika a with bar on top equals open parentheses table row 3 row cell negative 4 end cell end table close parentheses dan b with bar on top equals open parentheses table row 4 row cell negative 3 end cell end table close parentheses maka open vertical bar b with bar on top close vertical bar a with bar on top plus open vertical bar a with bar on top close vertical bar b with bar on top adalah ....space 

Pembahasan Soal:

Diketahui:

  • vektor a with bar on top equals open parentheses table row 3 row cell negative 4 end cell end table close parentheses 
  • vektor b with bar on top equals open parentheses table row 4 row cell negative 3 end cell end table close parentheses 

Ditanya: open vertical bar b with bar on top close vertical bar a with bar on top plus open vertical bar a with bar on top close vertical bar b with bar on top equals ?  

Jawab:

Rumus mencari panjang vektor v with bar on top equals open parentheses table row cell v subscript 1 end cell row cell v subscript 2 end cell end table close parentheses adalah open vertical bar v with bar on top close vertical bar equals square root of v subscript 1 squared plus v subscript 2 squared end root.

Rumus penjumlahan vektor u with bar on top equals open parentheses table row cell u subscript 1 end cell row cell u subscript 2 end cell end table close parentheses dan v with bar on top equals open parentheses table row cell v subscript 1 end cell row cell v subscript 2 end cell end table close parentheses adalah u with bar on top plus v with bar on top equals open parentheses table row cell u subscript 1 plus v subscript 1 end cell row cell u subscript 2 plus v subscript 2 end cell end table close parentheses.

Rumus perkalian vektor v with bar on top equals open parentheses table row cell v subscript 1 end cell row cell v subscript 2 end cell end table close parentheses dengan konstanta k adalah k v with bar on top equals open parentheses table row cell k times v subscript 1 end cell row cell k times v subscript 2 end cell end table close parentheses.

Dengan demikian, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar b with bar on top close vertical bar a with bar on top plus open vertical bar a with bar on top close vertical bar b with bar on top end cell equals cell square root of 4 squared plus open parentheses negative 3 close parentheses squared end root open parentheses table row 3 row cell negative 4 end cell end table close parentheses plus square root of 3 squared plus open parentheses negative 4 close parentheses squared end root open parentheses table row 4 row cell negative 3 end cell end table close parentheses end cell row blank equals cell square root of 25 open parentheses table row 3 row cell negative 4 end cell end table close parentheses plus square root of 25 open parentheses table row 4 row cell negative 3 end cell end table close parentheses end cell row blank equals cell 5 open parentheses table row 3 row cell negative 4 end cell end table close parentheses plus 5 open parentheses table row 4 row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 15 row cell negative 20 end cell end table close parentheses plus open parentheses table row 20 row cell negative 15 end cell end table close parentheses end cell row blank equals cell open parentheses table row 35 row cell negative 35 end cell end table close parentheses end cell end table 

Jadi,  open vertical bar b with bar on top close vertical bar a with bar on top plus open vertical bar a with bar on top close vertical bar b with bar on top adalah open parentheses table row 35 row cell negative 35 end cell end table close parentheses.space 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Ridha

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Jika a=(13​), b=(01​) dan c=(25​), maka panjang vektor d=3a−2b+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 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 u=4i+2j​ dan v=i−2j​. Panjang vektor 2u+v=....

Pembahasan Soal:

Jika diketahui vektor u with rightwards arrow on top equals open parentheses table row x row y end table close parentheses, maka dapat ditentukan panjang vektor u with rightwards arrow on top sebagai berikut.

open vertical bar u with rightwards arrow on top close vertical bar equals square root of x squared plus y squared end root

Jika diketahui vektor u with rightwards arrow on top equals open parentheses table row x row y end table close parentheses dan v with rightwards arrow on top equals open parentheses table row a row b end table close parentheses, serta k element of R, maka dapat ditentukan operasi hitung sebagai berikut.

k times u with rightwards arrow on top equals k times open parentheses table row x row y end table close parentheses equals open parentheses table row cell k x end cell row cell k y end cell end table close parentheses

u with rightwards arrow on top plus v with rightwards arrow on top equals open parentheses table row x row y end table close parentheses plus open parentheses table row a row b end table close parentheses equals open parentheses table row cell x plus a end cell row cell y plus b end cell end table close parentheses

Dengan menerapkan operasi penjumlahan dua vektor dan perkalian skalar dengan vektor, diperoleh perhitungan sebagai berikut.

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 end cell equals cell 2 open parentheses 4 i with hat on top plus 2 j with hat on top close parentheses plus open parentheses i with hat on top minus 2 j with hat on top close parentheses end cell row blank equals cell open parentheses 8 i with hat on top plus 4 j with hat on top close parentheses plus open parentheses i with hat on top minus 2 j with hat on top close parentheses end cell row blank equals cell 9 i with hat on top plus 2 j with hat on top end cell end table end style 

Dengan menerapkan rumus panjang vektor pada dua dimensi, diperoleh perhitungan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar 2 u with rightwards arrow on top plus v with rightwards arrow on top close vertical bar end cell equals cell square root of 9 squared plus 2 squared end root end cell row blank equals cell square root of 81 plus 4 end root end cell row blank equals cell square root of 85 end cell end table end style 

Dengan demikian, panjang vektor begin mathsize 14px style 2 u with rightwards arrow on top plus v with rightwards arrow on top end style adalah begin mathsize 14px style square root of 85 end style satuan.

1

Roboguru

Diketahui a=(4−1​) dan b=(12​) dan c=(−34​). Vektor yang panjangnya 3 dan searah dengan a−2b+2c  adalah ...

Pembahasan Soal:

Diketahui a with bar on top equals open parentheses table row 4 row cell negative 1 end cell end table close parentheses dan b with rightwards arrow on top equals open parentheses table row 1 row 2 end table close parentheses dan c with bar on top equals open parentheses table row cell negative 3 end cell row 4 end table close parentheses

Ingat!

  • Secara aljabar, perkalian vektor dengan skalar di R squared dapat dirumuskan dengan 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 maka k 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 a subscript 1 end cell row cell k a subscript 2 end cell end table close parentheses.
  • Pada vektor, jika 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 a with rightwards arrow on top minus b with rightwards arrow on top 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.
  • Pada vektor, jika 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 a with rightwards arrow on top plus b with rightwards arrow on top 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.
  • Jika 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 atau a with rightwards arrow on top equals left parenthesis a subscript 1 comma space a subscript 2 right parenthesis maka panjang vektor a with rightwards arrow on top adalah open vertical bar a with rightwards arrow on top close vertical bar equals square root of a subscript 1 squared plus a subscript 2 squared end root.

Sehingga:

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

Pada opsi, vektor yang panjangnya 3 satuan panjang dan searah dengan a with rightwards arrow on top minus 2 b with rightwards arrow on top plus 2 c with rightwards arrow on top adalah begin mathsize 12px style open parentheses table row cell negative 2 2 over 5 end cell row cell 1 4 over 5 end cell end table close parentheses end style. Karena:

  • Panjang vektor 3 satuan panjang

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell panjang space vektor space open parentheses table row cell negative 2 2 over 5 end cell row cell 1 4 over 5 end cell end table close parentheses space end cell equals cell square root of open parentheses negative 2 2 over 5 close parentheses squared plus open parentheses 1 4 over 5 close parentheses squared end root end cell row blank equals cell square root of open parentheses negative 12 over 5 close parentheses squared plus open parentheses 9 over 5 close parentheses squared end root end cell row blank equals cell square root of 144 over 25 plus 81 over 25 end root end cell row blank equals cell square root of 225 over 25 end root end cell row blank equals cell square root of 9 end cell row blank equals cell 3 space satuan space panjang end cell end table end style  

  • Searah dengan a with bar on top minus 2 b with bar on top plus 2 c with bar on top

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

Jadi, jawaban yang tepat adalah A.

0

Roboguru

Diketahui vektor u=(−4−2​) dan v=(−15​). Jika w​=​2u+3v​, nilai ∣∣​w∣∣​ adalah ...

Pembahasan Soal:

Berdasarkan konsep penjumlahan vektor pada dua dimensi maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight w with rightwards arrow on top end cell equals cell 2 straight u with rightwards arrow on top plus 3 straight v with rightwards arrow on top end cell row blank equals cell 2 open parentheses table row cell negative 4 end cell row cell negative 2 end cell end table close parentheses plus 3 open parentheses table row cell negative 1 end cell row 5 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 8 end cell row cell negative 4 end cell end table close parentheses plus open parentheses table row cell negative 3 end cell row 15 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 11 end cell row 11 end table close parentheses end cell end table 

sehingga nilai open vertical bar straight w with rightwards arrow on top close vertical bar adalah panjang vektor straight w with rightwards arrow on top maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight w with rightwards arrow on top end cell equals cell square root of open parentheses negative 11 close parentheses squared plus 11 squared end root end cell row blank equals cell square root of 121 plus 121 end root end cell row blank equals cell square root of 121 cross times 2 end root end cell row blank equals cell 11 square root of 2 end cell end table 

Oleh karena itu, jawaban yang benar adalah E.

0

Roboguru

Jika u=(−32​) dan v=(5−4​), nilai ∣∣​3u+v∣∣​ adalah ...

Pembahasan Soal:

Tentukan vektor 3 straight u with rightwards arrow on top plus straight v with rightwards arrow on top terlebih dahulu.

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 straight u with rightwards arrow on top plus straight v with rightwards arrow on top end cell equals cell 3 open parentheses table row cell negative 3 end cell row 2 end table close parentheses plus open parentheses table row 5 row cell negative 4 end cell end table close parentheses space end cell row blank equals cell open parentheses table row cell negative 9 end cell row 6 end table close parentheses plus open parentheses table row 5 row cell negative 4 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 4 end cell row 2 end table close parentheses end cell end table

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar 3 straight u with rightwards arrow on top plus straight v with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 4 close parentheses squared plus 2 squared end root end cell row blank equals cell square root of 16 plus 4 end root end cell row blank equals cell square root of 20 end cell row blank equals cell 2 square root of 5 end cell end table

Nilai open vertical bar 3 straight u with rightwards arrow on top plus straight v with rightwards arrow on top close vertical bar equals 2 square root of 5.

Jadi, pilihan jawaban yang tepat adalah A.

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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