Roboguru

Jika ∣∣​u∣∣​=6dan∣∣​v∣∣​=4 dan sudut antara udanv adalah 60∘, tentukanlah (u+v)⋅v!

Pertanyaan

Jika open vertical bar u with rightwards arrow on top close vertical bar equals 6 space dan space open vertical bar v with rightwards arrow on top close vertical bar equals 4 dan sudut antara u with rightwards arrow on top space dan space v with rightwards arrow on top adalah 60 degree, tentukanlah open parentheses u with rightwards arrow on top plus v with rightwards arrow on top close parentheses times v with rightwards arrow on top!

Pembahasan Soal:

Diketahui:

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 6 row cell open vertical bar v with rightwards arrow on top close vertical bar end cell equals 4 row theta equals cell 60 degree end cell end table 

maka dapat ditentukan:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses u with rightwards arrow on top plus v with rightwards arrow on top close parentheses times v with rightwards arrow on top end cell equals cell u with rightwards arrow on top times v with rightwards arrow on top plus v with rightwards arrow on top times v with rightwards arrow on top end cell row blank equals cell 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 space theta plus open vertical bar v with rightwards arrow on top close vertical bar squared end cell row blank equals cell 6 times 4 times cos space 60 degree plus 4 squared end cell row blank equals cell 24 times 1 half plus 16 end cell row blank equals cell 12 plus 16 end cell row blank equals 28 end table 

Dengan demikian, diperoleh table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses u with rightwards arrow on top plus v with rightwards arrow on top close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell v 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 28 end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

P. Afrisno

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Jika nilai ∣∣​a∣∣​=4, ∣∣​b∣∣​=6, dan ∣∣​a+b∣∣​=8, nilai ∣∣​a−b∣∣​ adalah....

Pembahasan Soal:

Diketahui:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals 4 row cell open vertical bar b with rightwards arrow on top close vertical bar end cell equals 6 row cell open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar end cell equals 8 end table

Sehingga

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 squared end cell equals cell 8 squared end cell row cell open vertical bar a 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 cos space alpha plus open vertical bar b with rightwards arrow on top close vertical bar end cell equals 64 row cell 4 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 cos space alpha plus 6 squared end cell equals 64 row cell 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 cos space alpha end cell equals cell 64 minus 16 minus 36 end cell row cell 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 cos space alpha end cell equals 12 end table

Selanjutnya

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 squared end cell equals cell open vertical bar a 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 cos space alpha plus open vertical bar b with rightwards arrow on top close vertical bar end cell row blank equals cell 4 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 cos space alpha plus 6 squared end cell row blank equals cell 16 minus 12 plus 36 end cell row blank equals 40 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 plus-or-minus square root of 40 end cell row blank equals cell plus-or-minus 2 square root of 10 end cell end table

Karena panjang vektor selalu positif maka nilai table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open vertical bar a with rightwards arrow on top minus 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 equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 10 end cell end table.

Oleh karena itu, jawaban yang benar adalah D.

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Jika 2 vektor v=(2a,3a,−1) dan w=(4,a,3) saling tegak lurus maka ada dua nilai a yang memenuhi, jumlah kedua nilai  tersebut adalah ...

Pembahasan Soal:

Syarat kedua vektor saling tegak lurus adalah straight v times straight w equals 0 atau hasil perkalian titik (dot product) adalah nol.

Diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight v times straight w end cell equals 0 row cell open parentheses 2 straight a comma space 3 straight a comma space minus 1 close parentheses open parentheses 4 comma space straight a comma space 3 close parentheses end cell equals 0 row cell open parentheses 2 straight a open parentheses 4 close parentheses close parentheses plus open parentheses 3 straight a open parentheses straight a close parentheses close parentheses plus open parentheses negative 1 open parentheses 3 close parentheses close parentheses end cell equals 0 row cell 8 straight a plus 3 straight a squared plus open parentheses negative 3 close parentheses end cell equals 0 row cell 8 straight a plus 3 straight a squared minus 3 end cell equals 0 row cell 3 straight a squared plus 8 straight a minus 3 end cell equals 0 row cell 3 straight a squared plus 9 straight a minus straight a minus 3 end cell equals 0 row cell 3 straight a open parentheses straight a plus 3 close parentheses minus open parentheses straight a plus 3 close parentheses end cell equals 0 row cell open parentheses 3 straight a minus 1 close parentheses open parentheses straight a plus 3 close parentheses end cell equals 0 end table

Kemudian diperoleh nilai a, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 straight a minus 1 end cell equals 0 row cell 3 straight a end cell equals 1 row straight a equals cell 1 third end cell row blank blank blank row cell straight a plus 3 end cell equals 0 row straight a equals cell negative 3 end cell end table

Jumlah kedua nilai a, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 third plus open parentheses negative 3 close parentheses end cell equals cell 1 third minus 3 end cell row blank equals cell 1 third minus 9 over 3 end cell row blank equals cell negative 8 over 3 end cell end table

Jadi, jawaban yang tepat adalah C.

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Jika a dan b adalah sembarang vektor dari ∣a∣ menyatakan panjang vektor , maka (a+b)∙(a+b)=…

Pembahasan Soal:

Ingat kembali rumus berikut.

  • top enclose a bullet top enclose a equals open vertical bar top enclose a close vertical bar squared 
  • top enclose a bullet top enclose b equals open vertical bar top enclose a close vertical bar times open vertical bar top enclose b close vertical bar times cos space alpha 
  • top enclose a bullet top enclose b equals top enclose b bullet top enclose a
  • table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses a with bar on top plus b with bar on top close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bullet end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses a with bar on top plus b with bar on top close parentheses 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 a with bar on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bullet end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell a with bar on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell a with bar on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bullet end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell b with bar on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell b with bar on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bullet end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell a with bar on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell b with bar on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bullet end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell b with bar on top end cell end table
  • table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar p with bar on top plus q with bar on top close vertical bar squared end cell equals cell open vertical bar p with bar on top close vertical bar squared plus open vertical bar q with bar on top close vertical bar squared plus 2 open vertical bar space p with bar on top close vertical bar times open vertical bar stack space q with bar on top close vertical bar times cos space alpha end cell end table  

Dari rumus diatas diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses a with bar on top plus b with bar on top close parentheses bullet open parentheses a with bar on top plus b with bar on top close parentheses end cell equals cell a with bar on top bullet a with bar on top plus a with bar on top bullet b with bar on top plus b with bar on top bullet a with bar on top plus b with bar on top bullet b with bar on top end cell row blank equals cell open vertical bar a with bar on top close vertical bar squared plus 2 open parentheses a with bar on top bullet b with bar on top close parentheses plus open vertical bar b with bar on top close vertical bar squared end cell row blank equals cell open vertical bar a with bar on top close vertical bar squared plus 2 times open vertical bar a with bar on top close vertical bar times open vertical bar b with bar on top close vertical bar times cos space alpha plus open vertical bar b with bar on top close vertical bar squared end cell row blank equals cell open vertical bar a with bar on top close vertical bar squared plus open vertical bar b with bar on top close vertical bar squared plus 2 times stack open vertical bar a close vertical bar with bar on top times open vertical bar b with bar on top close vertical bar times cos space alpha end cell row blank equals cell open vertical bar a with bar on top plus b with bar on top close vertical bar squared end cell end table   

Dengan demikian table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses a with bar on top plus b with bar on top close parentheses bullet open parentheses a with bar on top plus b with bar on top close parentheses end cell equals cell open vertical bar a with bar on top plus b with bar on top close vertical bar squared end cell end table

Jadi, jawaban yang benar adalah A.

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Jika vektor udanv saling tegak lurus, maka nilai dari u⋅v adalah ....

Pembahasan Soal:

Ingat bahwa jika vektor begin mathsize 14px style a with rightwards arrow on top space d a n space b with rightwards arrow on top end stylesaling tegak lurus, maka begin mathsize 14px style a with rightwards arrow on top times b with rightwards arrow on top equals 0 end style.
Oleh karena itu, jika vektor begin mathsize 14px style u with rightwards arrow on top space d a n space v with rightwards arrow on top end style saling tegak lurus, maka begin mathsize 14px style u with rightwards arrow on top times v with rightwards arrow on top end style = 0.

Jadi, jawaban yang tepat adalah C.

 

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Roboguru

Diketahui vektor   tegak lurus dengan vektor . Nilai dari n adalah ....

Pembahasan Soal:

Karena vektor begin mathsize 14px style a with rightwards arrow on top end styletegak lurus dengan vektor begin mathsize 14px style b with rightwards arrow on top end style maka

begin mathsize 14px style 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 0 row cell open parentheses table row 6 row 5 row cell negative 4 end cell end table close parentheses times open parentheses table row 3 row cell negative 2 end cell row n end table close parentheses end cell equals 0 row cell 6 times 3 plus 5 times open parentheses negative 2 close parentheses plus open parentheses negative 4 close parentheses times n end cell equals 0 row cell 18 minus 10 minus 4 n end cell equals 0 row cell 8 minus 4 n end cell equals 0 row cell negative 4 n end cell equals cell negative 8 end cell row n equals 2 end table end style

Jadi, jawaban yang tepat adalah D.

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Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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