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Tentukan besar sudut antara vektor  dan  dari pasangan vektor  dan  !

Pertanyaan

Tentukan besar sudut antara vektor a with rightwards arrow on top dan b with rightwards arrow on top dari pasangan vektor a with rightwards arrow on top equals open parentheses table row 3 row 1 row 2 end table close parentheses dan b with rightwards arrow on top equals open parentheses table row 3 row cell negative 5 end cell row cell negative 2 end cell end table close parentheses !

Pembahasan Soal:

Besar sudut antara dua vektor dapat dihitung dengan rumus berikut,

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos space theta end cell row blank equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar times open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator open parentheses table row 3 row 1 row 2 end table close parentheses times open parentheses table row 3 row cell negative 5 end cell row cell negative 2 end cell end table close parentheses over denominator open parentheses square root of 3 squared plus 1 squared plus 2 squared end root close parentheses open parentheses square root of 3 squared plus left parenthesis negative 5 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root close parentheses end fraction end cell row blank equals cell fraction numerator 9 minus 5 minus 4 over denominator square root of 9 plus 1 plus 4 end root square root of 9 plus 25 plus 4 end root end fraction end cell row blank equals cell fraction numerator 0 over denominator square root of 14 square root of 38 end fraction end cell row blank equals 0 end table 

Didapatkan nilai cos space theta equals 0 maka sudut theta equals 90 degree.

Jadi, besar sudut antara vektor a with rightwards arrow on top dan b with rightwards arrow on top adalah 90 degree.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Kumaralalita

Mahasiswa/Alumni Universitas Gadjah Mada

Terakhir diupdate 01 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui vektor-vektor , , dan . Jika  tegak lurus , hasil dari  = ...

Pembahasan Soal:

Jika begin mathsize 12px style a with rightwards harpoon with barb upwards on top perpendicular b with rightwards harpoon with barb upwards on top end style maka begin mathsize 12px style cos theta end style = 0. Dengan demikian, berlaku: begin mathsize 12px style a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top equals open parentheses a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3 close parentheses end style

Diketahui:

begin mathsize 12px style a with rightwards harpoon with barb upwards on top equals open parentheses table row 1 row 2 row cell negative 3 end cell end table close parentheses end stylebegin mathsize 12px style b with rightwards harpoon with barb upwards on top equals open parentheses table row 4 row 4 row m end table close parentheses end stylebegin mathsize 12px style c with rightwards harpoon with barb upwards on top equals open parentheses table row 3 row cell negative 4 end cell row 5 end table close parentheses end style

Karena begin mathsize 12px style a with rightwards harpoon with barb upwards on top perpendicular b with rightwards harpoon with barb upwards on top end style maka berlaku:

begin mathsize 12px style a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top equals 0 end style

begin mathsize 12px style left right double arrow 1 times 4 plus 2 times 4 minus 3 times m equals 0  left right double arrow 4 plus 8 minus 3 m equals 0  left right double arrow 3 m equals 12  left right double arrow m equals 4 end style

Maka:

begin mathsize 12px style a with rightwards harpoon with barb upwards on top plus 2 b with rightwards harpoon with barb upwards on top minus c with rightwards harpoon with barb upwards on top  equals open parentheses table row 1 row 2 row cell negative 3 end cell end table close parentheses plus 2 open parentheses table row 4 row 4 row 4 end table close parentheses minus open parentheses table row 3 row cell negative 4 end cell row 5 end table close parentheses  equals open parentheses table row 1 row 2 row cell negative 3 end cell end table close parentheses plus open parentheses table row 8 row 8 row 8 end table close parentheses minus open parentheses table row 3 row cell negative 4 end cell row 5 end table close parentheses  equals open parentheses table row cell 1 plus 8 minus 3 end cell row cell 2 plus 8 plus 4 end cell row cell negative 3 plus 8 minus 5 end cell end table close parentheses  equals open parentheses table row 6 row 14 row 0 end table close parentheses end style

Jadi, begin mathsize 12px style a with rightwards harpoon with barb upwards on top plus 2 b with rightwards harpoon with barb upwards on top minus c with rightwards harpoon with barb upwards on top end style = begin mathsize 12px style open parentheses table row 6 row 14 row 0 end table close parentheses end style.

0

Roboguru

Diketahui . Sudut antara vektor  dan  adalah ....

Pembahasan Soal:

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.

0

Roboguru

Tentukan hasil kali vektor-vektor berikut. a. , dan kedua vektor membentuk sudut .

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight p with rightwards arrow on top times straight q with rightwards arrow on top end cell equals cell open vertical bar straight p with rightwards arrow on top close vertical bar times open vertical bar straight q with rightwards arrow on top close vertical bar times cos space straight theta end cell row blank equals cell 8 times 10 times cos space 90 degree end cell row blank equals cell 80 times 0 end cell row blank equals 0 end table end style 

0

Roboguru

Tentukan hasil kali vektor-vektor berikut. d.

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight p with rightwards arrow on top times straight q with rightwards arrow on top end cell equals cell open parentheses table row 12 row cell negative 2 end cell end table close parentheses times open parentheses table row 8 row 23 end table close parentheses end cell row blank equals cell open parentheses 12 cross times 8 close parentheses plus open parentheses negative 2 cross times 23 close parentheses end cell row blank equals cell 96 plus open parentheses negative 46 close parentheses end cell row blank equals 50 end table end style     

0

Roboguru

Tentukan hasil kali vektor-vektor berikut. b.  dan kedua vektor membentuk sudut .

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight F subscript 1 times straight F subscript 2 end cell equals cell vertical line straight F subscript 1 vertical line times vertical line straight F subscript 2 vertical line times cos space straight theta end cell row blank equals cell 6 times 20 times cos space 180 degree end cell row blank equals cell 120 times open parentheses negative 1 close parentheses end cell row blank equals cell negative 120 end cell end table end style   

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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