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Diketahui vektor dan . Hitunglah , , dan !

Pertanyaan

Diketahui vektor begin mathsize 14px style a with rightwards arrow on top equals open parentheses table row 6 row 2 end table close parentheses space end styledan begin mathsize 14px style b with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 9 end table close parentheses end style. Hitunglah begin mathsize 14px style a with rightwards arrow on top times b with rightwards arrow on top end stylebegin mathsize 14px style open vertical bar a with rightwards arrow on top close vertical bar end style, dan begin mathsize 14px style open vertical bar b with rightwards arrow on top close vertical bar end style!

Pembahasan Soal:

Dot product dari begin mathsize 14px style a with rightwards arrow on top times b with rightwards arrow on top end style adalah

begin mathsize 14px style a with rightwards arrow on top times b with rightwards arrow on top equals open parentheses table row 6 row 2 end table close parentheses times open parentheses table row cell negative 3 end cell row 9 end table close parentheses equals negative 18 plus 18 equals 0 end style 

besar vektor begin mathsize 14px style a with rightwards arrow on top end style 

begin mathsize 14px style open vertical bar a with rightwards arrow on top close vertical bar equals square root of 6 squared plus 2 squared end root space space space space space space equals square root of 36 plus 4 end root space space space space space space equals square root of 40 space space space space space space equals 2 square root of 10 end style 

besar vektor begin mathsize 14px style b with rightwards arrow on top end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar b with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 3 close parentheses squared plus 9 squared end root end cell row cell space space space space space space end cell equals cell square root of 9 plus 81 end root end cell row cell space space space space space space end cell equals cell square root of 90 end cell row cell space space space space space space end cell equals cell 3 square root of 10 end cell end table end style 

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Roy

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 28 Maret 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Perhatikan gambar berikut!   Dengan aturan cosinus yaitu:   tunjukkan bahwa !

Pembahasan Soal:

Jika terdapat titik straight A open parentheses a subscript 1 comma space a subscript 2 close parentheses dan titik straight B open parentheses b subscript 1 comma space b subscript 2 close parentheses, maka berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell open parentheses a subscript 1 comma space a subscript 2 close parentheses comma space b with rightwards arrow on top equals open parentheses b subscript 1 comma space b subscript 2 close parentheses end cell row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell open parentheses a subscript 1 comma space a subscript 2 close parentheses times open parentheses b subscript 1 comma space b subscript 2 close parentheses end cell row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 end cell row cell AB with rightwards arrow on top end cell equals cell b with rightwards arrow on top minus a with rightwards arrow on top end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell square root of a subscript 1 squared plus a subscript 2 squared end root end cell end table  

Keterangan:

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 perkalian space titik space a with rightwards arrow on top space dan space b with rightwards arrow on top end cell row cell AB with rightwards arrow on top end cell equals cell vektor space posisi space dari space titik space straight B space ke space titik space straight A end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell panjang space a with rightwards arrow on top end cell end table 

Berdasarkan gambar, diketahui:

straight O open parentheses 0 comma space 0 close parentheses comma space straight P open parentheses a subscript 1 comma space a subscript 2 close parentheses comma space straight Q open parentheses b subscript 1 comma space b subscript 2 close parentheses space stack O P with rightwards arrow on top equals a with rightwards arrow on top stack O Q with rightwards arrow on top equals b with rightwards arrow on top  

Bukti:

begin mathsize 11px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar PQ with rightwards arrow on top close vertical bar squared end cell equals cell open vertical bar OP with rightwards arrow on top close vertical bar squared plus open vertical bar OQ with rightwards arrow on top close vertical bar squared minus 2 open vertical bar OP with rightwards arrow on top close vertical bar open vertical bar OQ with rightwards arrow on top close vertical bar cos space theta end cell row cell open parentheses square root of open parentheses b subscript 1 minus a subscript 1 close parentheses squared plus open parentheses b subscript 2 minus a subscript 2 close parentheses squared end root close parentheses squared end cell equals cell open parentheses square root of a subscript 1 squared plus a subscript 2 squared end root close parentheses squared plus open parentheses square root of b subscript 1 squared plus b subscript 2 squared end root close parentheses squared minus 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell open parentheses b subscript 1 minus a subscript 1 close parentheses squared plus open parentheses b subscript 2 minus a subscript 2 close parentheses squared end cell equals cell a subscript 1 squared plus a subscript 2 squared plus b subscript 1 squared plus b subscript 2 squared minus 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell b subscript 1 squared minus 2 a subscript 1 b subscript 1 plus a subscript 1 squared plus b subscript 2 squared minus 2 a subscript 2 b subscript 2 plus a subscript 2 squared end cell equals cell a subscript 1 squared plus a subscript 2 squared plus b subscript 1 squared plus b subscript 2 squared minus 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell negative 2 a subscript 1 b subscript 1 minus 2 a subscript 2 b subscript 2 end cell equals cell negative 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell negative 2 open parentheses a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 close parentheses end cell equals cell negative 2 times square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell row cell a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 end cell equals cell square root of a subscript 1 squared plus a subscript 2 squared end root times square root of b subscript 1 squared plus b subscript 2 squared end root times cos space theta end cell 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 cos space theta end cell end table end style 

Jadi, terbukti bahwa a with rightwards arrow on top times b with rightwards arrow on top equals 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 theta.

0

Roboguru

5. Jika   (1)   (2)   (3)    (4)   Pernyataan yang benar adalah...

Pembahasan Soal:

Perhatikan penghitungan berikut!

begin mathsize 14px style a with rightwards arrow on top equals open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses space semicolon space b with rightwards arrow on top equals open parentheses table row cell negative 4 end cell row 2 row cell negative 6 end cell end table close parentheses  left parenthesis 1 right parenthesis space open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar a with rightwards arrow on top plus b with rightwards arrow on top equals open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses plus open parentheses table row cell negative 4 end cell row 2 row cell negative 6 end cell end table close parentheses equals open parentheses table row cell negative 2 end cell row 1 row cell negative 3 end cell end table close parentheses open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar equals square root of open parentheses negative 2 close parentheses squared plus 1 squared plus open parentheses negative 3 close parentheses squared end root equals square root of 14   left parenthesis 2 right parenthesis space open vertical bar a with rightwards arrow on top close vertical bar colon open vertical bar b with rightwards arrow on top close vertical bar open vertical bar a with rightwards arrow on top close vertical bar equals square root of 2 squared plus open parentheses negative 1 close parentheses squared plus 3 squared end root equals square root of 14 open vertical bar b with rightwards arrow on top close vertical bar equals square root of open parentheses negative 4 close parentheses squared plus 2 squared plus open parentheses negative 6 close parentheses squared end root equals square root of 56 open vertical bar a with rightwards arrow on top close vertical bar colon open vertical bar b with rightwards arrow on top close vertical bar equals square root of 14 colon square root of 56 equals 1 colon 4  left parenthesis 3 right parenthesis space a with rightwards arrow on top bullet b with rightwards arrow on top open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses bullet open parentheses table row cell negative 4 end cell row 2 row cell negative 6 end cell end table close parentheses equals 2 open parentheses negative 4 close parentheses plus 2 open parentheses negative 1 close parentheses plus 3 open parentheses negative 6 close parentheses equals negative 28  left parenthesis 4 right parenthesis space a with rightwards arrow on top divided by divided by b with rightwards arrow on top a with rightwards arrow on top equals open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses space minus 2 a with rightwards arrow on top equals negative 2 open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses space equals open parentheses table row cell negative 4 end cell row 2 row cell negative 6 end cell end table close parentheses equals b with rightwards arrow on top vektor space a with rightwards arrow on top space jika space diperbesar space 2 space satuan space maka space akan space menjadi space vektor space b with rightwards arrow on top comma sehingga space vektor space a with rightwards arrow on top divided by divided by b with rightwards arrow on top  end style 

Pernyataan yang benar adalah (1),(3), dan (4). Jika disesuaikan dengan pilihan jawaban yang tersedia maka penyataan yang sesuai yaitu (1) dan (3).

Oleh karena itu, jawaban yang benar adalah B. undefined 

0

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.

0

Roboguru

Diketahui vektor satuan . Jika vektor  tegak lurus vektor  maka  sama dengan ...

Pembahasan Soal:

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 2 end cell row cell b subscript 1 end cell end table close parentheses, 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

Jika vektor a with rightwards arrow on top tegak lurus dengan 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 90 degree end cell row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals 0 end table

Panjang vektor a with rightwards arrow on top, yaitu 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

Vektor satuan adalah vektor dengan panjang satu satuan.

Apabila diketahui vektor satuan u with rightwards arrow on top equals a i with rightwards arrow on top plus 3 over 5 j with rightwards arrow on top, maka nilai a dapat ditentukan sebagai berikut.

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 a squared plus open parentheses 3 over 5 close parentheses squared end root end cell row 1 equals cell square root of a squared plus 9 over 25 end root end cell row 1 equals cell a squared plus 9 over 25 end cell row 25 equals cell 25 a squared plus 9 end cell row 0 equals cell 25 a squared minus 16 end cell row 0 equals cell open parentheses 5 a plus 4 close parentheses open parentheses 5 a minus 4 close parentheses end cell end table

a equals negative 4 over 5 space text atau end text space a equals 4 over 5

Berdasarkan konsep di atas, apabila vektor v with rightwards arrow on top tegak lurus u with rightwards arrow on top, maka dapat ditentukan hubungan berikut.

Untuk a equals negative 4 over 5

table attributes columnalign right center left columnspacing 0px end attributes row cell v with rightwards arrow on top times u with rightwards arrow on top end cell equals 0 row cell open parentheses table row 1 row b end table close parentheses open parentheses table row cell negative 4 over 5 end cell row cell 3 over 5 end cell end table close parentheses end cell equals 0 row cell negative 4 over 5 plus 3 over 5 b end cell equals 0 row cell 3 over 5 b end cell equals cell 4 over 5 end cell row cell 3 b end cell equals 4 row b equals cell 4 over 3 end cell end table

Diperoleh nilai a times b equals negative 4 over 5 times 4 over 3 equals negative 16 over 15

Untuk a equals 4 over 5

table attributes columnalign right center left columnspacing 0px end attributes row cell v with rightwards arrow on top times u with rightwards arrow on top end cell equals 0 row cell open parentheses table row 1 row b end table close parentheses open parentheses table row cell 4 over 5 end cell row cell 3 over 5 end cell end table close parentheses end cell equals 0 row cell 4 over 5 plus 3 over 5 b end cell equals 0 row cell 3 over 5 b end cell equals cell negative 4 over 5 end cell row b equals cell negative 4 over 3 end cell end table

Diperoleh nilai a times b equals 4 over 5 times open parentheses negative 4 over 3 close parentheses equals negative 16 over 15

Oleh karena itu, tidak terdapat jawaban yang tepat.

0

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.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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