Roboguru

Jika c merupakan vektor proyeksi a=2i−6j​+4k pada b=2i−j​+2k, maka panjang vektor  adalah ....

Pertanyaan

Jika begin mathsize 14px style straight c with rightwards arrow on top end style merupakan vektor proyeksi begin mathsize 14px style straight a with rightwards arrow on top equals stack 2 straight i with rightwards arrow on top minus 6 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top end style pada begin mathsize 14px style straight b with rightwards arrow on top equals 2 straight i with rightwards arrow on top minus straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top end style, maka panjang vektor begin mathsize 14px style straight c with rightwards arrow on top end style adalah ....

  1. 4

  2. 5

  3. 6

  4. 7

  5. 8

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight c with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator straight a with rightwards arrow on top. straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses stack 2 straight i with rightwards arrow on top minus 6 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top close parentheses space. space open parentheses 2 straight i with rightwards arrow on top minus straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses over denominator square root of 2 cubed plus open parentheses negative 1 close parentheses squared plus open parentheses 2 close parentheses squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 4 plus 6 plus 8 over denominator square root of 4 plus 1 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 18 over denominator square root of 9 end fraction close vertical bar end cell row blank equals cell open vertical bar 18 over 3 close vertical bar end cell row blank equals 6 end table end style 

Pembahasan terverifikasi oleh Roboguru

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A. Acfreelance

Terakhir diupdate 05 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui vektor p​=5i−7j​+4k pada  q​=i−2j​+2k. Jika vektor r adalah proyeksi dari vektor p​ pada q​, maka = ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight r with rightwards arrow on top close vertical bar end cell equals cell fraction numerator straight p with rightwards arrow on top. straight q with rightwards arrow on top over denominator open vertical bar straight q with rightwards arrow on top close vertical bar squared end fraction straight q with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses 5 straight i with rightwards arrow on top minus 7 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top close parentheses space. space open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses over denominator open parentheses square root of 2 squared plus open parentheses negative 1 close parentheses squared plus open parentheses 2 close parentheses squared end root close parentheses squared end fraction open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell fraction numerator 5 plus 14 plus 8 over denominator open parentheses square root of 9 close parentheses squared end fraction open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell 27 over 9 open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell 3 open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell 3 straight i with rightwards arrow on top minus 6 straight j with rightwards arrow on top plus 6 straight k with rightwards arrow on top end cell end table end style 

1

Roboguru

Diketahui x1​ dan x2​ adalah nilai-nilai x yang menyebabkan panjang proyeksi u=⎝⎛​1x5​⎠⎞​ terhadap v=⎝⎛​122​⎠⎞​ sama dengan 5.  Jika x1​<x2​, maka banyaknya bilangan bulat p yang memenuhi x1​<p≤...

Pembahasan Soal:

Diketahui bahwa begin mathsize 14px style u with rightwards arrow on top equals open parentheses table row 1 row x row 5 end table close parentheses end style dan begin mathsize 14px style v with rightwards arrow on top equals open parentheses table row 1 row 2 row 2 end table close parentheses end style. Kemudian, panjang proyeksi begin mathsize 14px style u with rightwards arrow on top end style terhadap begin mathsize 14px style v with rightwards arrow on top end style sama dengan 5. Akibatnya, didapat persamaan berikut.

begin mathsize 14px style open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar equals 5 end style 

Dapat diperhatikan bahwa begin mathsize 14px style u with rightwards arrow on top times v with rightwards arrow on top end style dapat dihitung sebagai berikut.

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 open parentheses table row 1 row x row 5 end table close parentheses times open parentheses table row 1 row 2 row 2 end table close parentheses end cell row blank equals cell 1 left parenthesis 1 right parenthesis plus x left parenthesis 2 right parenthesis plus 5 left parenthesis 2 right parenthesis space end cell row blank equals cell 1 plus 2 x plus 10 space end cell row blank equals cell 2 x plus 11 end cell end table

Kemudian, nilai dari begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell v with rightwards arrow on top end cell end table end style adalah sebagai berikut.

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 1 squared plus 2 squared plus 2 squared end root end cell row blank equals cell square root of 1 plus 4 plus 4 end root end cell row blank equals cell square root of 9 end cell row blank equals 3 end table end style 

Kemudian, substitusi nilai begin mathsize 14px style u with rightwards arrow on top times v with rightwards arrow on top equals 11 plus 2 x end style dan begin mathsize 14px style open vertical bar v with rightwards arrow on top close vertical bar equals 3 end style ke persamaan begin mathsize 14px style open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar equals 5 end style.

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar end cell equals 5 row cell open vertical bar fraction numerator 2 x plus 11 over denominator 3 end fraction close vertical bar end cell equals 5 row cell fraction numerator vertical line 2 x plus 11 vertical line over denominator vertical line 3 vertical line end fraction end cell equals 5 row cell fraction numerator vertical line 2 x plus 11 vertical line over denominator 3 end fraction end cell equals 5 row cell vertical line 2 x plus 11 vertical line end cell equals 15 end table

Akibatnya, didapat 2 x plus 11 equals 15 atau 2 x plus 11 equals negative 15.

Kemudian, perhatikan perhitungan berikut!

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x plus 11 end cell equals cell 15 space end cell row cell 2 x end cell equals cell 15 minus 11 space end cell row cell 2 x end cell equals 4 row x equals 2 end table 

atau

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x plus 11 end cell equals cell negative 15 space end cell row cell 2 x end cell equals cell negative 15 minus 11 space end cell row cell 2 x end cell equals cell negative 26 space end cell row x equals cell negative 13 end cell end table 

Didapat begin mathsize 14px style x equals 2 end style atau begin mathsize 14px style x equals negative 13 end style.

Pada soal, diketahui bahwa begin mathsize 14px style x subscript 1 less than x subscript 2 comma end style maka begin mathsize 14px style straight x subscript 1 equals negative 13 end style dan begin mathsize 14px style x subscript 2 equals 2. end style  

Oleh karena itu, bilangan bulat begin mathsize 14px style p end style yang memenuhi begin mathsize 14px style negative 13 less than p less or equal than 2 end style adalah begin mathsize 14px style negative 12 end stylebegin mathsize 14px style negative 11 end stylebegin mathsize 14px style negative 10 end stylebegin mathsize 14px style negative 9 end stylebegin mathsize 14px style negative 8 end stylebegin mathsize 14px style negative 7 end styleundefinedbegin mathsize 14px style negative 5 end stylebegin mathsize 14px style negative 4 end stylebegin mathsize 14px style negative 3 end styleundefinedbegin mathsize 14px style negative 1 end stylebegin mathsize 14px style 0 end stylebegin mathsize 14px style 1 end style.

Dengan demikian, banyak bilangan ulat begin mathsize 14px style p end style yang memenuhi negative 13 less than p less than 2 adalah 14.

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Bila panjang proyeksi skalar vektor b=i−2j​ pada vektor a=xi+yj​ dengan x,y > 0 adalah 1, maka nilai 4x - 3y + 1 =…

Pembahasan Soal:

begin mathsize 14px style fraction numerator straight b with rightwards arrow on top space. space straight a with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar end fraction equals 1 fraction numerator open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses space. space open parentheses table row straight x row straight y row 0 end table close parentheses over denominator square root of straight x squared plus straight y squared end root end fraction equals 1 straight x minus 2 straight y equals square root of straight x squared plus straight y squared end root open parentheses straight x minus 2 straight y close parentheses squared equals straight x squared plus straight y squared straight x squared plus 4 straight y squared minus 4 xy equals straight x squared plus straight y squared 3 straight y squared minus 4 xy equals 0 straight y open parentheses 3 straight y minus 4 straight x close parentheses equals 0 end style 

DIdapat 3y - 4x = 0 atau y = 0

Karena y > 0, maka dipilih 3y - 4x = 0.

begin mathsize 14px style 4 straight x minus 3 straight y plus 1 equals 1 minus left parenthesis 3 straight y minus 4 straight x right parenthesis space equals 1 minus 0 space equals 1 space end style

0

Roboguru

Diketahui vektor a=−i−j​+2k dan b=i−j​−2k. Proyeksi vektor ortogonal a pada b adalah ....

Pembahasan Soal:

Diketahui vektor begin mathsize 14px style a with rightwards arrow on top equals negative i with rightwards arrow on top minus j with rightwards arrow on top plus 2 k with rightwards arrow on top end style dan begin mathsize 14px style b with rightwards arrow on top equals i with rightwards arrow on top minus j with rightwards arrow on top minus 2 k with rightwards arrow on top end style

Misal begin mathsize 14px style c with rightwards arrow on top end style adalah proyeksi vektor ortogonal dari begin mathsize 14px style a with rightwards arrow on top end style pada begin mathsize 14px style b with rightwards arrow on top end style, maka kita dapat perhitungan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell open parentheses fraction numerator a with rightwards arrow on top space. space b with rightwards arrow on top over denominator open vertical bar space b with rightwards arrow on top close vertical bar squared end fraction close parentheses b with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator left parenthesis negative i with rightwards arrow on top minus j with rightwards arrow on top plus 2 k with rightwards arrow on top right parenthesis. space left parenthesis i with rightwards arrow on top minus j with rightwards arrow on top minus 2 k with rightwards arrow on top right parenthesis over denominator open parentheses square root of 1 squared plus open parentheses negative 1 close parentheses squared plus open parentheses negative 2 close parentheses squared end root close parentheses squared end fraction close parentheses left parenthesis i with rightwards arrow on top minus j with rightwards arrow on top minus 2 k with rightwards arrow on top right parenthesis end cell row blank equals cell open parentheses fraction numerator negative 1 plus 1 minus 4 over denominator 6 end fraction close parentheses left parenthesis i with rightwards arrow on top minus j with rightwards arrow on top minus 2 k with rightwards arrow on top right parenthesis end cell row blank equals cell negative 4 over 6 left parenthesis i with rightwards arrow on top minus j with rightwards arrow on top minus 2 k with rightwards arrow on top right parenthesis end cell row blank equals cell negative 2 over 3 left parenthesis i with rightwards arrow on top minus j with rightwards arrow on top minus 2 k with rightwards arrow on top right parenthesis end cell row blank equals cell negative 2 over 3 i with rightwards arrow on top plus 2 over 3 j with rightwards arrow on top plus 4 over 3 k with rightwards arrow on top end cell end table end style

Jadi, jawabannya adalah E.

0

Roboguru

Vektor w merupakan proyeksi vektor u=3i−2j​+xk terhadap vektor v=2i+4j​+4k. Jika ∣∣​w∣∣​=3, hasil penjumlahan semua nilai x yang mungkin adalah ....

Pembahasan Soal:

Karena vektor begin mathsize 14px style w with rightwards arrow on top end style merupakan proyeksi vektor begin mathsize 14px style u with rightwards arrow on top equals 3 i with rightwards arrow on top minus 2 j with rightwards arrow on top plus x k with rightwards arrow on top end style terhadap vektor begin mathsize 14px style v with rightwards arrow on top equals 2 i with rightwards arrow on top plus 4 j with rightwards arrow on top plus 4 k with rightwards arrow on top comma end style maka dapat ditulis hubungan sebagai berikut.

begin mathsize 14px style w with rightwards arrow on top equals open parentheses fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction close parentheses space v with rightwards arrow on top end style

Perhatikan perhitungan di bawah ini!

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 3 left parenthesis 2 right parenthesis plus left parenthesis negative 2 right parenthesis left parenthesis 4 right parenthesis plus x left parenthesis 4 right parenthesis end cell row blank equals cell 6 minus 8 plus 4 x space end cell row blank equals cell 4 x minus 2 end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar v with rightwards arrow on top close vertical bar end cell equals cell square root of 2 squared plus 4 squared plus 4 squared end root end cell row blank equals cell square root of 4 plus 16 plus 16 end root end cell row blank equals cell square root of 36 end cell row blank equals 6 end table end style

Oleh karena itu, didapat vektor begin mathsize 14px style w with rightwards arrow on top end style adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell size 14px w with size 14px rightwards arrow on top end cell size 14px equals cell begin mathsize 14px style open parentheses fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction close parentheses end style size 14px v with size 14px rightwards arrow on top end cell row cell size 14px w with size 14px rightwards arrow on top end cell size 14px equals cell begin mathsize 14px style open parentheses fraction numerator 4 x minus 2 over denominator 6 squared end fraction close parentheses end style size 14px v with size 14px rightwards arrow on top end cell row cell size 14px w with size 14px rightwards arrow on top end cell size 14px equals cell begin mathsize 14px style open parentheses fraction numerator 4 x minus 2 over denominator 36 end fraction close parentheses end style size 14px v with size 14px rightwards arrow on top end cell end table

Misal begin mathsize 14px style p equals fraction numerator 4 x minus 2 over denominator 36 end fraction comma end style maka didapat begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell w 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 p 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 end style.

Karena begin mathsize 14px style open vertical bar w with rightwards arrow on top close vertical bar equals 3 comma end style maka diperoleh nilai begin mathsize 14px style p end style dari perhitungan di bawah ini.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar w with rightwards arrow on top close vertical bar end cell equals 3 row cell open vertical bar p v with rightwards arrow on top close vertical bar end cell equals 3 row cell open vertical bar p close vertical bar open vertical bar v with rightwards arrow on top close vertical bar end cell equals 3 row cell open vertical bar p close vertical bar times 6 end cell equals 3 row cell open vertical bar p close vertical bar end cell equals cell 1 half end cell row p equals cell 1 half space atau space p equals negative 1 half end cell end table end style

Untuk begin mathsize 14px style p equals 1 half comma end style maka didapat nilai undefined sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row p equals cell 1 half end cell row cell fraction numerator 4 x minus 2 over denominator 36 end fraction end cell equals cell 1 half end cell row cell 4 x minus 2 end cell equals 18 row cell 4 x end cell equals 20 row x equals 5 end table end style

Untuk begin mathsize 14px style p equals negative 1 half comma end style maka didapat nilai undefined sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row p equals cell negative 1 half end cell row cell fraction numerator 4 x minus 2 over denominator 36 end fraction end cell equals cell negative 1 half end cell row cell 4 x minus 2 end cell equals cell negative 18 end cell row cell 4 x end cell equals cell negative 16 end cell row x equals cell negative 4 end cell end table end style

Dengan demikian, didapat size 14px x size 14px equals size 14px 5 atau begin mathsize 14px style x equals negative 4 end style.

Oleh karena itu, hasil penjumlahan semua nilai undefined yang mungkin adalah begin mathsize 14px style 5 plus left parenthesis negative 4 right parenthesis equals 1 end style.

Jadi, jawaban yang tepat adalah B.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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