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Diketahui titik A(2, 7, 8), B(−1, 1, −1), dan C(0, 3, 2). Jika u mewakili AB dan v mewakili BC, maka proyeksi ortogonal vektor  pada  adalah ....

Pertanyaan

Diketahui titik straight A open parentheses 2 comma space 7 comma space 8 close parenthesesstraight B open parentheses negative 1 comma space 1 comma space minus 1 close parentheses, dan straight C open parentheses 0 comma space 3 comma space 2 close parentheses. Jika u with rightwards arrow on top mewakili AB with rightwards arrow on top dan v with rightwards arrow on top mewakili BC with rightwards arrow on top, maka proyeksi ortogonal vektor u with rightwards arrow on top pada v with rightwards arrow on top adalah ....

  1. negative 3 i with rightwards arrow on top minus 6 j with rightwards arrow on top minus 9 k with rightwards arrow on top

  2. i with rightwards arrow on top plus 2 j with rightwards arrow on top plus 3 k with rightwards arrow on top

  3. 1 third i with rightwards arrow on top plus 2 over 3 j with rightwards arrow on top plus k with rightwards arrow on top

  4. negative 9 i with rightwards arrow on top minus 18 j with rightwards arrow on top minus 27 k with rightwards arrow on top

  5. 3 i with rightwards arrow on top plus 6 j with rightwards arrow on top plus 9 k with rightwards arrow on top

Y. Fathoni

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Gunakan konsep proyeksi orthogonal vektor u with rightwards arrow on top pada v with rightwards arrow on top.

Proyeksi space ortogonal 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 v with rightwards arrow on top

Diketahui:
Titik straight A open parentheses 2 comma space 7 comma space 8 close parenthesesstraight B open parentheses negative 1 comma space 1 comma space minus 1 close parentheses, dan straight C open parentheses 0 comma space 3 comma space 2 close parentheses. Jika u with rightwards arrow on top mewakili AB with rightwards arrow on top dan v with rightwards arrow on top mewakili BC with rightwards arrow on top.
Akan ditentukan proyeksi ortogonal vektor u with rightwards arrow on top pada v with rightwards arrow on top.

Tentukan vektor u with rightwards arrow on top.

table attributes columnalign right center left columnspacing 2px end attributes row cell u with rightwards arrow on top end cell equals cell AB with rightwards arrow on top end cell row blank equals cell straight B minus straight A end cell row blank equals cell open parentheses negative 1 minus 2 comma space 1 minus 7 comma space minus 1 minus 8 close parentheses end cell row cell u with rightwards arrow on top end cell equals cell open parentheses negative 3 comma space minus 6 comma space minus 9 close parentheses end cell end table

Tentukan vektor v with rightwards arrow on top.

table attributes columnalign right center left columnspacing 2px end attributes row cell v with rightwards arrow on top end cell equals cell BC with rightwards arrow on top end cell row blank equals cell straight C minus straight B end cell row blank equals cell open parentheses 0 minus open parentheses negative 1 close parentheses comma space 3 minus 1 comma space 2 minus open parentheses negative 1 close parentheses close parentheses end cell row cell v with rightwards arrow on top end cell equals cell open parentheses 1 comma space 2 comma space 3 close parentheses end cell end table

Sehingga proyeksi ortogonal vektor u with rightwards arrow on top pada v with rightwards arrow on top dapat diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 2px end attributes row Proyeksi equals cell 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 times v with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator open parentheses negative 3 close parentheses times 1 plus open parentheses negative 6 close parentheses times 2 plus open parentheses negative 9 close parentheses times 3 over denominator open parentheses square root of 1 squared plus 2 squared plus 3 squared end root close parentheses squared end fraction close parentheses open parentheses 1 comma space 2 comma space 3 close parentheses end cell row blank equals cell open parentheses fraction numerator negative 3 minus 12 minus 27 over denominator open parentheses square root of 1 plus 4 plus 9 end root close parentheses squared end fraction close parentheses open parentheses 1 comma space 2 comma space 3 close parentheses end cell row blank equals cell open parentheses 42 over open parentheses square root of 14 close parentheses squared close parentheses open parentheses 1 comma space 2 comma space 3 close parentheses end cell row blank equals cell open parentheses 42 over 14 close parentheses open parentheses 1 comma space 2 comma space 3 close parentheses end cell row blank equals cell 3 cross times open parentheses 1 comma space 2 comma space 3 close parentheses end cell row Proyeksi equals cell open parentheses 3 comma space 6 comma space 9 close parentheses end cell end table

Sehingga diperoleh proyeksi ortogonalnya adalah open parentheses 3 comma space 6 comma space 9 close parentheses atau 3 i with rightwards arrow on top plus 6 j with rightwards arrow on top plus 9 k with rightwards arrow on top.

Jadi, jawaban yang tepat adalah E.

90

5.0 (3 rating)

Pertanyaan serupa

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Jawaban terverifikasi

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