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Jika a=(13​), b=(01​) dan c=(25​), maka panjang vektor d=3a−2b+c adalah ...

Pertanyaan

Jika straight a with rightwards arrow on top equals open parentheses table row 1 row 3 end table close parentheses, straight b with rightwards arrow on top equals open parentheses table row 0 row 1 end table close parentheses dan straight c with rightwards arrow on top equals open parentheses table row 2 row 5 end table close parentheses, maka 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 adalah ...

  1. 11

  2. 12

  3. 13

  4. 14

  5. 15

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Nuryani

Mahasiswa/Alumni Universitas Padjadjaran

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

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 a=2p​+q​,b=p​−4q​ dan c=mp​+(m+n−1)q​ dengan m dan n konstanta serta vektor p​ dan q​ dua vektor tidak saling sejajar. Jika c=2a−2b. Nilai (m+n) adalah ...

Pembahasan Soal:

Diketahui a with rightwards arrow on top equals 2 p with rightwards arrow on top plus q with rightwards arrow on top dan b with rightwards arrow on top equals p with rightwards arrow on top minus 4 q with rightwards arrow on top, maka 

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 a with rightwards arrow on top minus 2 b with rightwards arrow on top end cell equals cell 2 open parentheses 2 p with rightwards arrow on top plus q with rightwards arrow on top close parentheses minus 2 open parentheses p with rightwards arrow on top minus 4 q with rightwards arrow on top close parentheses end cell row blank equals cell open parentheses 4 p with rightwards arrow on top plus 2 q with rightwards arrow on top close parentheses minus open parentheses 2 p with rightwards arrow on top minus 8 q with rightwards arrow on top close parentheses end cell row blank equals cell 4 p with rightwards arrow on top minus 2 p with rightwards arrow on top plus 2 q with rightwards arrow on top plus 8 q with rightwards arrow on top end cell row blank equals cell 2 p with rightwards arrow on top plus 10 q with rightwards arrow on top space end cell end table

Diketahui pula c with rightwards arrow on top equals m p with rightwards arrow on top plus open parentheses m plus n minus 1 close parentheses q with rightwards arrow on top dan c with rightwards arrow on top equals 2 a with rightwards arrow on top minus 2 b with rightwards arrow on top, maka 

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell 2 a with rightwards arrow on top minus 2 b with rightwards arrow on top end cell row cell m p with rightwards arrow on top plus open parentheses m plus n minus 1 close parentheses q with rightwards arrow on top end cell equals cell 2 p with rightwards arrow on top plus 10 q with rightwards arrow on top space end cell row blank blank blank end table 

Dari kesamaan vektor tersebut diperoleh 

m equals 2 space dan space m plus n minus 1 equals 10 

Substitusi m equals 2 ke m plus n minus 1 equals 10, diperoleh 

table attributes columnalign right center left columnspacing 0px end attributes row cell m plus n minus 1 end cell equals 10 row cell 2 plus n minus 1 end cell equals 10 row cell 1 plus n end cell equals 10 row n equals 9 end table 

Sehingga 

table attributes columnalign right center left columnspacing 0px end attributes row cell m plus n end cell equals cell 2 plus 9 end cell row blank equals 11 end table 

Jadi, nilai open parentheses m plus n close parentheses adalah 9

0

Roboguru

Diketahui koordinat titik K(2,−1), L(−5,3), dan M(−1,−5). Tentukan: a. vektor KL⇀ dan KM⇀ b. hasil KL⇀+KM⇀ c. hasil 3KL⇀−2KM⇀

Pembahasan Soal:

Berdasarkan koordinat titik di atas, dapat diperoleh

a. begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell KL with rightwards harpoon with barb upwards on top end cell equals cell OL with rightwards harpoon with barb upwards on top minus OK with rightwards harpoon with barb upwards on top end cell row blank equals cell open parentheses negative 5 comma space 3 close parentheses minus open parentheses 2 comma space minus 1 close parentheses end cell row blank equals cell open parentheses negative 5 minus 2 comma space 3 minus left parenthesis negative 1 right parenthesis close parentheses end cell row blank equals cell left parenthesis negative 7 comma space 4 right parenthesis end cell end table end style dan begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell KM with rightwards harpoon with barb upwards on top end cell equals cell OM with rightwards harpoon with barb upwards on top minus OK with rightwards harpoon with barb upwards on top end cell row blank equals cell left parenthesis negative 1 comma space minus 5 right parenthesis minus left parenthesis 2 comma space minus 1 right parenthesis end cell row blank equals cell left parenthesis negative 1 minus 2 comma space minus 5 minus left parenthesis negative 1 right parenthesis right parenthesis end cell row blank equals cell left parenthesis negative 3 comma space minus 4 right parenthesis end cell end table end style 

 

b. begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell KL with rightwards harpoon with barb upwards on top plus KM with rightwards harpoon with barb upwards on top end cell equals cell open parentheses negative 7 comma space 4 close parentheses plus open parentheses negative 3 comma space minus 4 close parentheses end cell row blank equals cell open parentheses negative 7 plus left parenthesis negative 3 right parenthesis comma space 4 plus left parenthesis negative 4 right parenthesis close parentheses end cell row blank equals cell open parentheses negative 10 comma space 0 close parentheses end cell end table end style 

 

c. begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 3 KL with rightwards harpoon with barb upwards on top minus 2 KM with rightwards harpoon with barb upwards on top end cell equals cell 3 open parentheses negative 7 comma space 4 close parentheses minus 2 open parentheses negative 3 comma space minus 4 close parentheses end cell row blank equals cell open parentheses negative 21 comma space 12 close parentheses minus open parentheses negative 6 comma space minus 8 close parentheses end cell row blank equals cell open parentheses negative 21 minus left parenthesis negative 6 right parenthesis comma space 12 minus left parenthesis negative 8 right parenthesis close parentheses end cell row blank equals cell open parentheses negative 15 comma space 20 close parentheses end cell end table end style 

0

Roboguru

Diketahui vektor a=(3−2​), vektor b=(41​), dan vektor c=(−2−1​). Jika 2a−3b+kc=(0−4​), dengan k 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 koordinat titik A(4,−3), B(−1,−5) dan C(−2,3). Jika a⇀, b⇀ dan c⇀ berturut-turut vektor posisi titik A, B, dan C, hasil 2a⇀−3b⇀+c⇀ adalah …

Pembahasan Soal:

Vektor posisi titik A, B, dan C yaitu:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards harpoon with barb upwards on top end cell equals cell 4 i with hat on top minus 3 j with hat on top end cell row cell b with rightwards harpoon with barb upwards on top end cell equals cell negative i with hat on top minus 5 j with hat on top end cell row cell c with rightwards harpoon with barb upwards on top end cell equals cell negative 2 i with hat on top plus 3 j with hat on top end cell end table end style 

Dengan konsep perkalian vektor dengan skalar diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 a with rightwards harpoon with barb upwards on top end cell equals cell 2 open parentheses 4 i with hat on top minus 3 j with hat on top close parentheses end cell row blank equals cell 8 i with hat on top minus 6 j with hat on top end cell row cell 3 b with rightwards harpoon with barb upwards on top end cell equals cell 3 open parentheses negative i with hat on top minus 5 j with hat on top close parentheses end cell row blank equals cell negative 3 i with hat on top minus 15 j with hat on top end cell end table end style

Operasi penjumlahan dan pengurangan vektor secara aljabar dari begin mathsize 14px style 2 a with rightwards harpoon with barb upwards on top minus 3 b with rightwards harpoon with barb upwards on top plus c with rightwards harpoon with barb upwards on top end style yaitu:

2a3b+c====(8i6j)(3i15j)+(2i+3j)8i6j+3i+15j2i+3j(8+32)i+(6+15+3)j9i+12j

 

Jadi, hasil begin mathsize 14px style 2 a with rightwards harpoon with barb upwards on top minus 3 b with rightwards harpoon with barb upwards on top plus c with rightwards harpoon with barb upwards on top end style adalah begin mathsize 14px style 9 i with hat on top plus 12 j with hat on top 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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