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Diketahui ∣ ∣ ​ a ∣ ∣ ​ = 6 , ∣ ∣ ​ a + b ∣ ∣ ​ = 2 19 ​ , dan ∣ ∣ ​ a − b ∣ ∣ ​ = 2 7 ​ . Sudut antara vektor a dan b adalah ....

Diketahui . Sudut antara vektor  dan  adalah ....

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H. Endah

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Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

sudut antara vektor dan adalah .

 sudut antara vektor a with rightwards arrow on top dan b with rightwards arrow on top adalah 60 degree.

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Pembahasan

Diketahui , maka didapatkan: Diketahui , maka didapatkan: Eliminasi Persamaan dan dan didapatkan: Karena tidak mungkin negatif, maka diambil . Substitusikan ke Persamaan dan didapatkan: Jadi,sudut antara vektor dan adalah .

Diketahui open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar equals 2 square root of 19, maka didapatkan:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar end cell equals cell 2 square root of 19 end cell row cell square root of open vertical bar a with rightwards arrow on top close vertical bar squared plus open vertical bar b with rightwards arrow on top close vertical bar squared plus 2 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 root end cell equals cell 2 square root of 19 end cell row cell 6 squared plus open vertical bar b with rightwards arrow on top close vertical bar squared plus 2 open parentheses 6 close parentheses open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals 76 row cell 36 plus open vertical bar b with rightwards arrow on top close vertical bar squared plus 12 open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals 76 row cell open vertical bar b with rightwards arrow on top close vertical bar squared plus 12 open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals cell 76 minus 36 end cell row cell open vertical bar b with rightwards arrow on top close vertical bar squared plus 12 open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals cell 40 space horizontal ellipsis space open parentheses 1 close parentheses end cell end table  

Diketahui open vertical bar a with rightwards arrow on top minus b with rightwards arrow on top close vertical bar equals 2 square root of 7, maka didapatkan:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a with rightwards arrow on top minus b with rightwards arrow on top close vertical bar end cell equals cell 2 square root of 7 end cell row cell square root of open vertical bar a with rightwards arrow on top close vertical bar squared plus open vertical bar b with rightwards arrow on top close vertical bar squared minus 2 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 root end cell equals cell 2 square root of 7 end cell row cell 6 squared plus open vertical bar b with rightwards arrow on top close vertical bar squared minus 2 open parentheses 6 close parentheses open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals 28 row cell 36 plus open vertical bar b with rightwards arrow on top close vertical bar squared minus 12 open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals 28 row cell open vertical bar b with rightwards arrow on top close vertical bar squared minus 12 open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals cell 28 minus 36 end cell row cell open vertical bar b with rightwards arrow on top close vertical bar squared minus 12 open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha end cell equals cell negative 8 space horizontal ellipsis space open parentheses 2 close parentheses end cell end table   

Eliminasi Persamaan open parentheses 1 close parentheses dan open parentheses 2 close parentheses dan didapatkan:

table row cell 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 squared plus table attributes columnalign right center left columnspacing 0px end attributes row blank blank 12 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 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 40 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table end cell space row cell bottom enclose 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 squared minus table attributes columnalign right center left columnspacing 0px end attributes row blank blank 12 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 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 negative 8 end cell end table end enclose end cell plus row cell space space space space space space space space space space space space space space space space space space 2 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 squared equals 32 end cell space row cell space space space space space space space space space space space space space space space space space space space space 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 squared equals 16 end cell space row cell space space space space space space space space space space space space space space space space space space space space space space 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 equals plus-or-minus 4 end cell space end table   

Karena open vertical bar b with rightwards arrow on top close vertical bar tidak mungkin negatif, maka diambil open vertical bar b with rightwards arrow on top close vertical bar equals 4.

Substitusikan open vertical bar b with rightwards arrow on top close vertical bar equals 4 ke Persamaan open parentheses 1 close parentheses dan didapatkan:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 4 close parentheses squared plus 12 open parentheses 4 close parentheses space cos space alpha end cell equals 40 row cell 16 plus 48 space cos space alpha end cell equals 40 row cell 48 space cos space alpha end cell equals cell 40 minus 16 end cell row cell 48 space cos space alpha end cell equals 24 row cell cos space alpha end cell equals cell 24 over 48 end cell row cell cos space alpha end cell equals cell 1 half end cell row alpha equals cell 60 degree end cell end table 

Jadi, sudut antara vektor a with rightwards arrow on top dan b with rightwards arrow on top adalah 60 degree.

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