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Diketahui vektor-vektor begin mathsize 12px style u with rightwards harpoon with barb upwards on top equals b i with rightwards harpoon with barb upwards on top plus a j with rightwards harpoon with barb upwards on top plus 9 k with rightwards harpoon with barb upwards on top end style dan begin mathsize 12px style v with rightwards harpoon with barb upwards on top equals a i with rightwards harpoon with barb upwards on top minus b j with rightwards harpoon with barb upwards on top plus a k with rightwards harpoon with barb upwards on top end style. Sudut antara vektor begin mathsize 12px style u with rightwards harpoon with barb upwards on top end style dan begin mathsize 12px style v with rightwards harpoon with barb upwards on top end style adalah begin mathsize 12px style theta end style dengan begin mathsize 12px style cos theta equals 6 over 11 end style. Proyeksi begin mathsize 12px style u with rightwards harpoon with barb upwards on top end style pada begin mathsize 12px style v with rightwards harpoon with barb upwards on top end style adalah begin mathsize 12px style p with rightwards harpoon with barb upwards on top equals 4 a with rightwards harpoon with barb upwards on top minus 2 j with rightwards harpoon with barb upwards on top plus 4 k with rightwards harpoon with barb upwards on top end style. Nilai dari b = ...

  1. begin mathsize 12px style square root of 2 end style

  2. 2

  3. 2begin mathsize 12px style square root of 2 end style

  4. 4

  5. 4begin mathsize 12px style square root of 2 end style

Y. Laksmi

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

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Jawaban

nilai b = 2begin mathsize 12px style square root of 2 end style

Pembahasan

Proyeksi vektor begin mathsize 12px style u with rightwards harpoon with barb upwards on top end style pada begin mathsize 12px style v with rightwards harpoon with barb upwards on top end style adalah vektor begin mathsize 12px style p with rightwards harpoon with barb upwards on top end style maka berlaku:

  • begin mathsize 12px style open vertical bar p with rightwards harpoon with barb upwards on top close vertical bar equals open vertical bar u with rightwards harpoon with barb upwards on top close vertical bar cos theta end style
  • begin mathsize 12px style cos theta equals fraction numerator u with rightwards harpoon with barb upwards on top times v with rightwards harpoon with barb upwards on top over denominator open vertical bar u with rightwards harpoon with barb upwards on top close vertical bar open vertical bar v with rightwards harpoon with barb upwards on top close vertical bar end fraction end style
  • Vektor begin mathsize 12px style p with rightwards harpoon with barb upwards on top end style sejajar dengan vektor begin mathsize 12px style v with rightwards harpoon with barb upwards on top end style maka begin mathsize 12px style p with rightwards harpoon with barb upwards on top equals n times v with rightwards harpoon with barb upwards on top end style

Diketahui:

  • begin mathsize 12px style u with rightwards harpoon with barb upwards on top equals b i with rightwards harpoon with barb upwards on top plus a j with rightwards harpoon with barb upwards on top plus 9 k with rightwards harpoon with barb upwards on top end style
  • begin mathsize 12px style v with rightwards harpoon with barb upwards on top equals a i with rightwards harpoon with barb upwards on top minus b j with rightwards harpoon with barb upwards on top plus a k with rightwards harpoon with barb upwards on top end style
  • begin mathsize 12px style p with rightwards harpoon with barb upwards on top equals 4 a with rightwards harpoon with barb upwards on top minus 2 j with rightwards harpoon with barb upwards on top plus 4 k with rightwards harpoon with barb upwards on top end style
  • begin mathsize 12px style cos theta equals 6 over 11 end style

Karena begin mathsize 12px style p with rightwards harpoon with barb upwards on top end style sejajar dengan vektor begin mathsize 12px style v with rightwards harpoon with barb upwards on top end style maka: 

begin mathsize 12px style n times v with rightwards harpoon with barb upwards on top equals p with rightwards harpoon with barb upwards on top  n equals open parentheses table row a row cell negative b end cell row a end table close parentheses equals open parentheses table row 4 row cell negative 2 end cell row 4 end table close parentheses end style

Dari persamaan di atas diperoleh:

na = 4begin mathsize 12px style left right double arrow end stylen = begin mathsize 12px style 4 over a end style    ...  (i)

-bn = -2begin mathsize 12px style left right double arrow end stylen = begin mathsize 12px style 2 over b end style  ...  (ii)

Dari persamaan (i) dan (ii) diperoleh:

begin mathsize 12px style 4 over a end stylebegin mathsize 12px style 2 over b end stylebegin mathsize 12px style left right double arrow end stylea = 2b ... (iii)

 

Sudut antara kedua vektor adalah begin mathsize 12px style theta end style sehingga berlaku:

begin mathsize 12px style cos theta equals fraction numerator u with rightwards harpoon with barb upwards on top times v with rightwards harpoon with barb upwards on top over denominator open vertical bar u with rightwards harpoon with barb upwards on top close vertical bar open vertical bar v with rightwards harpoon with barb upwards on top close vertical bar end fraction space  left right double arrow 6 over 11 equals fraction numerator a b minus a b plus 9 a over denominator square root of b squared plus a squared plus 81 end root times square root of a squared plus b squared plus a squared end root end fraction  left right double arrow 6 over 11 equals fraction numerator 9 a over denominator square root of b squared plus a squared plus 81 end root times square root of 2 a squared plus b squared end root end fraction space space... space left parenthesis i v right parenthesis end style

Substitusikan persamaan (iii) ke persamaan (iv), diiperoleh:

begin mathsize 12px style 6 over 11 equals fraction numerator 18 b over denominator square root of b squared plus 4 b squared plus 81 end root begin display style times end style begin display style square root of 8 b squared plus b squared end root end style end fraction  left right double arrow 6 over 11 equals fraction numerator 18 b over denominator square root of 5 b squared plus 81 end root begin display style times end style begin display style 3 end style begin display style b end style end fraction  left right double arrow 6 over 11 equals fraction numerator 6 over denominator square root of 5 b squared plus 81 end root end fraction  left right double arrow square root of 5 b squared plus 81 end root equals 11  left right double arrow 5 b squared plus 81 equals 121  left right double arrow 5 b squared equals 40  left right double arrow b squared equals 8  left right double arrow b equals plus-or-minus 2 square root of 2 end style

Jadi, nilai b = 2begin mathsize 12px style square root of 2 end style

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Diketahui vektor-vektor u=9i+aj​+bk dan v=ai−bj​+ak. Sudut antara vektor u dan vektor v adalah θ dengan cosθ=116​. Proyeksi  pada  adalah p​=4i−2j​+4k. Nilai b = ....

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