Roboguru

Diketahui  dan . Tentukanlah: a.

Pertanyaan

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

a. straight a with rightwards arrow on top plus straight b with rightwards arrow on top 

Pembahasan Soal:

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

straight a with rightwards arrow on top plus straight b with rightwards arrow on top equals open parentheses table row 3 row cell negative 2 end cell row 1 end table close parentheses plus open parentheses table row 1 row 3 row cell negative 2 end cell end table close parentheses equals open parentheses table row 4 row 1 row cell negative 1 end cell end table close parentheses 

Jadi, hasilnya adalah open parentheses table row 4 row 1 row cell negative 1 end cell end table close parentheses.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

W. Lestari

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui titik-titik: ,  dan . Jika  maka panjang vektor

Pembahasan Soal:

Diketahui titik-titik: straight A left parenthesis 5 comma space 2 right parenthesis, straight B left parenthesis 4 comma space 6 right parenthesis dan straight C left parenthesis 3 comma space 2 right parenthesis. Dapat ditulis kedalam vektor posisi menjadi stack O A with rightwards arrow on top equals open parentheses table row 5 row 2 end table close parenthesesstack O B with rightwards arrow on top equals open parentheses table row 4 row 6 end table close parentheses dan stack O C with rightwards arrow on top equals open parentheses table row 3 row 2 end table close parentheses.

Ingat!

  • Pada vektor, stack A B with rightwards arrow on top didefinisikan dengan stack A B with rightwards arrow on top equals stack O B with rightwards arrow on top minus stack O A with rightwards arrow on top equals b with rightwards arrow on top minus a with rightwards arrow on top.
  • 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 v with rightwards arrow on top end cell equals cell stack A B with rightwards arrow on top plus stack A C with rightwards arrow on top end cell row blank equals cell stack O B with rightwards arrow on top minus stack O A with rightwards arrow on top plus stack O C with rightwards arrow on top minus stack O A with rightwards arrow on top end cell row blank equals cell open parentheses table row 4 row 6 end table close parentheses minus open parentheses table row 5 row 2 end table close parentheses plus open parentheses table row 3 row 2 end table close parentheses minus open parentheses table row 5 row 2 end table close parentheses end cell row blank equals cell open parentheses table row cell 4 minus 5 end cell row cell 6 minus 2 end cell end table close parentheses plus open parentheses table row cell 3 minus 5 end cell row cell 2 minus 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell row 4 end table close parentheses plus open parentheses table row cell negative 2 end cell row 0 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 plus left parenthesis negative 2 right parenthesis end cell row cell 4 plus 0 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 3 end cell row 4 end table close parentheses end cell row blank blank blank row cell open vertical bar v with rightwards arrow on top close vertical bar end cell equals cell square root of left parenthesis negative 3 right parenthesis squared plus 4 to the power of 2 end exponent end root end cell row blank equals cell square root of 9 plus 16 end root end cell row blank equals cell square root of 25 end cell row blank equals cell 5 space satuan space panjang space end cell end table 

Jadi, jawaban yang tepat adalah D.

Roboguru

Diketahui vektor  dan . Tentukanlah  !

Pembahasan Soal:

Dari soal diketahui

a with rightwards arrow on top equals open parentheses table row 3 row cell negative 2 end cell row 1 end table close parentheses b with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 4 row 5 end table close parentheses c with rightwards arrow on top equals open parentheses table row 5 row cell negative 3 end cell row 2 end table close parentheses 

Maka

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top plus b with rightwards arrow on top plus 2 c with rightwards arrow on top end cell equals cell open parentheses table row 3 row cell negative 2 end cell row 1 end table close parentheses plus open parentheses table row cell negative 3 end cell row 4 row 5 end table close parentheses minus 2 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 0 row 2 row 6 end table close parentheses minus open parentheses table row 10 row cell negative 6 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 10 end cell row 8 row 2 end table close parentheses end cell end table

Jadi, hasil dari a with rightwards arrow on top plus b with rightwards arrow on top plus 2 c with rightwards arrow on top adalahopen parentheses table row cell negative 10 end cell row 8 row 2 end table close parentheses 

Roboguru

Diketahui , ,  tentukan !

Pembahasan Soal:

Gunakan konsep perkalian vektor dengan skalar, penjumlahan, dan pengurangan vektor.

Diketahui a with rightwards arrow on top equals open parentheses table row cell negative 2 end cell row 5 row cell negative 3 end cell end table close parenthesesb with rightwards arrow on top equals open parentheses table row 4 row cell negative 6 end cell row 4 end table close parenthesesc with rightwards arrow on top equals open parentheses table row cell negative 4 end cell row 11 row 3 end table close parentheses

Perhatikan perhitungan berikut.

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

Jadi, diperoleh 3 a with rightwards arrow on top plus 2 b with rightwards arrow on top equals open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses.

Roboguru

Diketahui vektor  tegak lurus dengan vektor . Jika  maka

Pembahasan Soal:

Jika vektor a with rightwards arrow on top equals a subscript 1 i with rightwards arrow on top plus a subscript 2 j with rightwards arrow on top plus a subscript 3 k with rightwards arrow on top dan b with rightwards arrow on top equals b subscript 1 i with rightwards arrow on top plus b subscript 2 j with rightwards arrow on top plus b subscript 3 k with rightwards arrow on top, maka

a with rightwards arrow on top times b with rightwards arrow on top equals a subscript 1 times b subscript 1 plus a subscript 2 times b subscript 2 plus a subscript 3 times b subscript 3

Jika vektor a with rightwards arrow on top tegak lurus vektor b with rightwards arrow on top, maka 

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals 0 end table

Berdasarkan konsep di atas, nilai x dapat ditentukan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals 0 row cell open parentheses table row 2 row 3 row x end table close parentheses open parentheses table row 3 row cell negative 6 end cell row 2 end table close parentheses end cell equals 0 row cell 2 times 3 plus 3 open parentheses negative 6 close parentheses plus 2 x end cell equals 0 row cell 6 minus 18 plus 2 x end cell equals 0 row cell negative 12 plus 2 x end cell equals 0 row cell 2 x end cell equals 12 row x equals 6 end table

Diperoleh vektor x equals 6 sehingga a with rightwards arrow on top equals 2 i with rightwards arrow on top plus 3 j with rightwards arrow on top plus 6 k with rightwards arrow on top dan c with rightwards arrow on top equals 6 i with rightwards arrow on top minus 5 j with rightwards arrow on top plus k with rightwards arrow on top. Dapat ditentukan operasi penjumlahan dan pengurangan vektor berikut.

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 plus c with rightwards arrow on top end cell equals cell open parentheses table row 2 row 3 row 6 end table close parentheses minus open parentheses table row 3 row cell negative 6 end cell row 2 end table close parentheses plus open parentheses table row 6 row cell negative 5 end cell row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 minus 3 plus 6 end cell row cell 3 minus open parentheses negative 6 close parentheses plus open parentheses negative 5 close parentheses end cell row cell 6 minus 2 plus 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row 5 row 4 row 5 end table close parentheses end cell end table

Diperoleh 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 minus 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 plus 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 5 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 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 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 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 5 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell k with rightwards arrow on top end cell end table

Oleh karena itu, jawaban yang tepat adalah B.

Roboguru

Diketahui  dan  dan . Vektor yang panjangnya 3 dan searah dengan   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.

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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

Membantu

Kurang Membantu

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