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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 ....

Pertanyaan

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. end style Jika begin mathsize 14px style open vertical bar w with rightwards arrow on top close vertical bar equals 3 comma end style hasil penjumlahan semua nilai begin mathsize 14px style x end style yang mungkin adalah ....

  1. begin mathsize 14px style 9 end style 

  2. begin mathsize 14px style 1 end style 

  3. begin mathsize 14px style negative 1 end style 

  4. begin mathsize 14px style negative 9 end style 

  5. begin mathsize 14px style negative 20 end style 

Pembahasan:

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.

Jawaban terverifikasi

Dijawab oleh:

S. Difhayanti

Mahasiswa/Alumni Universitas Muhammadiyah Prof. Dr. Hamka

Terakhir diupdate 14 Oktober 2021

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Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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