Roboguru

Diketahui  dengan koordinat  dan .  Tentukan:

Pertanyaan

Diketahui triangle ABC dengan koordinat straight A open parentheses 2 comma space minus 1 comma space minus 1 close parentheses comma space straight B open parentheses negative 1 comma space 4 comma space minus 2 close parentheses dan straight C open parentheses 5 comma space 0 comma space minus 3 close parentheses
Tentukan: 

  • AB with rightwards arrow on top times AC with rightwards arrow on top  

Pembahasan Soal:

Diketahui triangle ABC dengan koordinat straight A open parentheses 2 comma space minus 1 comma space minus 1 close parentheses comma space straight B open parentheses negative 1 comma space 4 comma space minus 2 close parentheses dan straight C open parentheses 5 comma space 0 comma space minus 3 close parentheses

Maka

table attributes columnalign right center left columnspacing 0px end attributes row cell AB with rightwards arrow on top end cell equals cell straight B minus straight A end cell row blank equals cell open parentheses negative 1 comma space 4 comma space minus 2 close parentheses minus open parentheses 2 comma space minus 1 comma space minus 1 close parentheses end cell row blank equals cell open parentheses negative 3 comma space 5 comma space minus 1 close parentheses end cell row blank blank blank row cell AC with rightwards arrow on top end cell equals cell straight C minus straight A end cell row blank equals cell open parentheses 5 comma space 0 comma space minus 3 close parentheses minus open parentheses 2 comma space minus 1 comma space minus 1 close parentheses end cell row blank equals cell open parentheses 3 comma space 1 comma space minus 2 close parentheses end cell end table 

Sehingga 

table attributes columnalign right center left columnspacing 0px end attributes row cell AB with rightwards arrow on top times AC with rightwards arrow on top end cell equals cell open parentheses negative 3 comma space 5 comma space minus 1 close parentheses times open parentheses 3 comma space 1 comma space minus 2 close parentheses end cell row blank equals cell open parentheses negative 9 close parentheses plus 5 plus 2 end cell row blank equals cell negative 2 end cell end table 

Jadi,  AB with rightwards arrow on top times AC with rightwards arrow on top equals negative 2 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

L. Rante

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 05 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui vektor ,  dan . Tentukan hasil dari b.

Pembahasan Soal:

Gunakan konsep perkalian dot (perkalian titik) secara aljabar, misalkan terdapat vektor begin mathsize 14px style 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 end style dan begin mathsize 14px style 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 end style berlaku perkalian yang langsung melibatkan unsur-unsur vektornya, yaitu

begin mathsize 14px style 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 end style

Diketahui:

begin mathsize 14px style p with rightwards arrow on top equals negative 4 i with rightwards arrow on top minus 7 j with rightwards arrow on top end stylebegin mathsize 14px style q with rightwards arrow on top equals negative i with rightwards arrow on top plus 3 j with rightwards arrow on top end style dan begin mathsize 14px style r with rightwards arrow on top equals 2 i with rightwards arrow on top plus j with rightwards arrow on top end style

Akan ditentukan hasil dari begin mathsize 14px style p with rightwards arrow on top times q with rightwards arrow on top end style.

Perhatikan perhitungan berikut.

begin mathsize 14px style p with rightwards arrow on top times q with rightwards arrow on top equals open parentheses negative 4 cross times negative 1 close parentheses plus open parentheses negative 7 cross times 3 close parentheses equals 5 minus 21 equals 16 end style 

Jadi, diperoleh nilai begin mathsize 14px style p with rightwards arrow on top times q with rightwards arrow on top equals 16 end style.

Roboguru

Sebuah vektor  dengan panjang  membuat sudut lancip dengan . Bila vektor  diproyeksikan ke vektor  panjang proyeksinya sama dengan . Vektor  tersebut adalah ...

Pembahasan Soal:

Panjang vektor a with rightwards arrow on top equals open parentheses table row x row y end table close parentheses, yaitu open vertical bar a with rightwards arrow on top close vertical bar equals square root of x squared plus y squared end root

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

Panjang proyeksi vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top dapat ditentukan menggunakan rumus berikut.

open vertical bar c with rightwards arrow on top close vertical bar equals fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction

Berdasarkan konsep di atas dapat ditentukan hubungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar c with rightwards arrow on top close vertical bar end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator open vertical bar y close vertical bar end fraction end cell row cell 4 over 5 square root of 10 end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator square root of 3 squared plus 1 squared end root end fraction end cell row cell 4 over 5 square root of 10 end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator square root of 10 end fraction end cell row cell x with rightwards arrow on top times y with rightwards arrow on top end cell equals cell fraction numerator 4 square root of 10 times square root of 10 over denominator 5 end fraction end cell row cell x with rightwards arrow on top times y with rightwards arrow on top end cell equals 8 end table

Misal vektor x with rightwards arrow on top equals open square brackets a comma space b close square brackets dapat ditentukan persamaan-persamaan berikut.

Persamaan 1:

table attributes columnalign right center left columnspacing 0px end attributes row cell x with rightwards arrow on top times y with rightwards arrow on top end cell equals 8 row cell open parentheses table row a row b end table close parentheses open parentheses table row 3 row 1 end table close parentheses end cell equals 8 row cell 3 a plus b end cell equals 8 row b equals cell 8 minus 3 a end cell end table

Persamaan 2:

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of a squared plus b squared end root end cell equals cell open vertical bar x with rightwards arrow on top close vertical bar end cell row cell square root of a squared plus b squared end root end cell equals cell 2 square root of 2 end cell row cell a squared plus b squared end cell equals 8 end table

Substitusi persamaan 1 ke persamaan 2:

table attributes columnalign right center left columnspacing 0px end attributes row cell a squared plus b squared end cell equals 8 row cell a squared plus open parentheses 8 minus 3 a close parentheses squared end cell equals 8 row cell a squared plus 64 minus 48 a plus 9 a squared end cell equals 8 row cell 10 a squared minus 48 a plus 56 end cell equals 0 row cell 5 a squared minus 24 a plus 28 end cell equals 0 row cell open parentheses a minus 2 close parentheses open parentheses 5 a minus 14 close parentheses end cell equals 0 end table

a equals 2 atau a equals 14 over 5

Untuk a equals 2 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row b equals cell 8 minus 3 a end cell row blank equals cell 8 minus 3 times 2 end cell row blank equals 2 end table

Untuk a equals 14 over 5 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row b equals cell 8 minus 3 a end cell row blank equals cell 8 minus 3 times 14 over 5 end cell row blank equals cell 40 over 5 minus 42 over 5 end cell row blank equals cell negative 2 over 5 end cell end table

Vektor x with rightwards arrow on top equals open square brackets 2 comma space 2 close square brackets space text atau end text space open square brackets 14 over 5 comma space minus 2 over 5 close square brackets

Oleh karena itu, jawaban yang tepat adalah A.

Roboguru

Diketahui vektor  dan . Tentukan sudut antara  dan .

Pembahasan Soal:

Jika vektor begin mathsize 14px style u with rightwards arrow on top equals open parentheses table row cell x subscript 1 end cell row cell y subscript 1 end cell row cell z subscript 1 end cell end table close parentheses end style dan begin mathsize 14px style v with rightwards arrow on top equals open parentheses table row cell x subscript 2 end cell row cell y subscript 2 end cell row cell z subscript 2 end cell end table close parentheses end style, maka hasil kali titik (dot product) dari dua vektor adalah

begin mathsize 14px style u with rightwards arrow on top times v with rightwards arrow on top equals x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 plus z subscript 1 z subscript 2 end style

dan

begin mathsize 14px style u with rightwards arrow on top times v with rightwards arrow on top equals open vertical bar u with rightwards arrow on top close vertical bar open vertical bar v with rightwards arrow on top close vertical bar cos theta end style

dengan begin mathsize 14px style theta end style adalah sudut di antara kedua vektor.

Perhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar u with rightwards arrow on top close vertical bar end cell equals cell square root of x subscript 1 squared plus y subscript 1 squared plus z subscript 1 squared end root end cell row blank equals cell square root of 2 squared plus open parentheses negative 1 close parentheses squared plus 1 squared end root end cell row blank equals cell square root of 4 plus 1 plus 1 end root end cell row blank equals cell square root of 6 end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell vertical line v with rightwards arrow on top vertical line end cell equals cell square root of x subscript 2 squared plus y subscript 2 squared plus z subscript 2 squared end root end cell row blank equals cell square root of 7 squared plus 0 squared plus 1 squared end root end cell row blank equals cell square root of 49 plus 0 plus 1 end root end cell row blank equals cell square root of 50 end cell row blank equals cell 5 square root of 2 end cell end table end style

Akibatnya

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell u with rightwards arrow on top times v with rightwards arrow on top end cell equals cell vertical line u with rightwards arrow on top vertical line vertical line v with rightwards arrow on top vertical line space cos space theta end cell row cell open parentheses table row 2 row cell negative 1 end cell row 1 end table close parentheses times open parentheses table row 7 row 0 row 1 end table close parentheses end cell equals cell open parentheses square root of 6 close parentheses open parentheses 5 square root of 2 close parentheses space cos space theta end cell row cell 14 plus 0 plus 1 end cell equals cell 5 square root of 12 space cos space theta end cell row 15 equals cell 10 square root of 3 space cos space theta end cell row cell fraction numerator 15 over denominator 10 square root of 3 end fraction end cell equals cell cos space theta end cell row cell fraction numerator 15 over denominator 10 square root of 3 end fraction times fraction numerator square root of 3 over denominator square root of 3 end fraction end cell equals cell cos space theta end cell row cell 15 over 30 square root of 3 end cell equals cell cos space theta end cell row cell 1 half square root of 3 end cell equals cell cos space theta end cell row cell 30 degree end cell equals theta end table end style

Jadi, besar sudut antara begin mathsize 14px style u with rightwards arrow on top end style dan begin mathsize 14px style v with rightwards arrow on top end style adalah begin mathsize 14px style 30 degree end style.

Roboguru

Diketahui 3 buah vektor yaitu: Jika , maka tentukan nilai .

Pembahasan Soal:

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 times b with rightwards arrow on top adalah:

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 parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell end table close parentheses times open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell end table close parentheses end cell row blank equals cell a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 end cell end table 

open vertical bar a with rightwards harpoon with barb upwards on top close vertical bar adalah panjang vektor a with rightwards arrow on top, dengan:

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

Diketahui:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell negative 8 i with rightwards arrow on top plus 4 j with rightwards arrow on top end cell row cell b with rightwards arrow on top end cell equals cell 3 i with rightwards arrow on top plus 8 j with rightwards arrow on top end cell row cell c with rightwards arrow on top end cell equals cell 4 i with rightwards arrow on top minus y j with rightwards arrow on top end cell end table

Jika a with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar c with rightwards arrow on top close vertical bar, maka tentukan nilai y.

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 c with rightwards arrow on top close vertical bar end cell row cell open parentheses table row cell negative 8 end cell row 4 end table close parentheses times open parentheses table row 3 row 8 end table close parentheses end cell equals cell square root of 4 squared plus open parentheses negative y close parentheses squared end root end cell row cell open parentheses negative 8 close parentheses times 3 plus 4 times 8 end cell equals cell square root of 16 plus y squared end root end cell row cell negative 24 plus 32 end cell equals cell square root of 16 plus y squared end root end cell row cell 8 squared end cell equals cell open parentheses square root of 16 plus y squared end root close parentheses squared end cell row 64 equals cell 16 plus y squared end cell row 48 equals cell y squared end cell row cell y squared end cell equals 48 row y equals cell plus-or-minus square root of 48 end cell row y equals cell plus-or-minus square root of 16 times 3 end root end cell row y equals cell plus-or-minus 4 square root of 3 end cell end table 

Jadi, table attributes columnalign right center left columnspacing 0px end attributes row blank blank y 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 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y 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 minus 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 square root of 3 end cell end table.

Roboguru

Sudut antara vektor  dan  sama dengan . Jika  dan  maka

Pembahasan Soal:

Diketahui:

  • alpha equals 150 degree (sudut antara vektor a with rightwards arrow on top dan b with rightwards arrow on top)
  • open vertical bar b with bar on top close vertical bar equals 6
  • a. b equals negative 9 

Ingat!

  • Perkalian skalar vektor a with rightwards arrow on top dengan vektor b with rightwards arrow on top open parentheses 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. 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 close parentheses didefinisikan sebagai hasil kali panjang vektor a with rightwards arrow on top dan panjang vektor b with rightwards arrow on top dengan kosinus sudut yang diapit oleh vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top atau 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. 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 equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell stack open vertical bar a close vertical bar with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open vertical bar b with rightwards arrow on top close vertical bar end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank alpha end table.
  • a with rightwards arrow on top. a with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar squared 

Maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top. 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 negative 9 end cell equals cell open vertical bar a with rightwards arrow on top close vertical bar cross times 6 cross times cos space 150 degree end cell row cell negative 9 end cell equals cell open vertical bar a with rightwards arrow on top close vertical bar cross times 6 cross times open parentheses negative 1 half square root of 3 close parentheses end cell row cell negative 9 end cell equals cell open vertical bar a with rightwards arrow on top close vertical bar cross times open parentheses negative 3 square root of 3 close parentheses end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell fraction numerator negative 9 over denominator negative 3 square root of 3 end fraction end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell fraction numerator 3 over denominator square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell square root of 3 space satuan space panjang end cell end table 

Sehingga:

 table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top. left parenthesis b with rightwards arrow on top minus a with rightwards arrow on top right parenthesis end cell equals cell a with rightwards arrow on top. b with rightwards arrow on top minus a with rightwards arrow on top. a with rightwards arrow on top end cell row blank equals cell a with rightwards arrow on top. b with rightwards arrow on top minus open vertical bar a with rightwards arrow on top close vertical bar squared end cell row blank equals cell negative 9 minus open parentheses square root of 3 close parentheses squared end cell row blank equals cell negative 9 minus 3 end cell row blank equals cell negative 12 end cell end table 

Jadi, jawaban yang tepat adalah B.

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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