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Diketahui vektor-vektor u=bi−12j​+ak dan v=ai+aj​−bk. Sudut antara vektor u dan v adalah θ dengan cosθ=43​​· Proyeksi vektor  pada adalah p​=−4i−4j​+4k. Nilai dari b = ...

Pertanyaan

Diketahui vektor-vektor begin mathsize 14px style u with rightwards arrow on top equals b i with rightwards arrow on top minus 12 j with rightwards arrow on top plus a k with rightwards arrow on top end style dan begin mathsize 14px style v with rightwards arrow on top equals a i with rightwards arrow on top plus a j with rightwards arrow on top minus b k with rightwards arrow on top end style. Sudut antara vektor begin mathsize 14px style u with rightwards arrow on top end style dan begin mathsize 14px style v with rightwards arrow on top end style adalah θ dengan begin mathsize 14px style cos space theta equals fraction numerator square root of 3 over denominator 4 end fraction end style· Proyeksi vektor begin mathsize 14px style u with rightwards arrow on top end style pada begin mathsize 14px style v with rightwards arrow on top end style adalah begin mathsize 14px style p with rightwards arrow on top equals negative 4 i with rightwards arrow on top minus 4 j with rightwards arrow on top plus 4 k with rightwards arrow on top end style. Nilai dari b = ...

  1. begin mathsize 14px style 4 square root of 7 end style

  2. begin mathsize 14px style 2 square root of 14 end style

  3. begin mathsize 14px style 2 square root of 7 end style

  4. begin mathsize 14px style square root of 14 end style

  5. begin mathsize 14px style square root of 7 end style

Pembahasan:

begin mathsize 14px style straight p with rightwards arrow on top space proyeksi space straight u with rightwards arrow on top space pada space straight v with rightwards arrow on top space maka space straight p with rightwards arrow on top space dan space straight v with rightwards arrow on top space searah comma space sehingga colon  straight p with rightwards arrow on top equals straight n. straight space straight v with rightwards arrow on top rightwards double arrow open parentheses table row cell negative 4 end cell row cell negative 4 end cell row 4 end table close parentheses equals straight n straight space. straight space open parentheses table row straight a row straight a row cell negative straight b end cell end table close parentheses    Jadi comma space minus 4 equals straight n left parenthesis straight a right parenthesis rightwards double arrow straight n equals negative 4 over straight a space dan space 4 equals negative nb rightwards double arrow straight n equals negative 4 over straight b  sehingga colon  minus 4 over straight a equals negative 4 over straight b rightwards double arrow straight a equals straight b    cos invisible function application space straight theta equals fraction numerator straight u with rightwards arrow on top straight space. straight space straight v with rightwards arrow on top over denominator open vertical bar straight u with rightwards arrow on top close vertical bar open vertical bar straight v with rightwards arrow on top close vertical bar end fraction  fraction numerator square root of 3 over denominator 4 end fraction equals fraction numerator ab minus 12 straight a minus ab over denominator square root of straight b squared plus open parentheses negative 12 close parentheses squared plus straight a squared end root. straight space square root of straight a squared plus straight a squared plus left parenthesis negative straight b right parenthesis squared end root end fraction  Substitusi space straight a equals straight b  fraction numerator square root of 3 over denominator 4 end fraction equals fraction numerator straight b squared minus 12 straight b minus straight b squared over denominator square root of straight b squared plus open parentheses negative 12 close parentheses squared plus straight b squared end root. straight space square root of straight b squared plus straight b squared plus left parenthesis negative straight b right parenthesis squared end root end fraction  fraction numerator square root of 3 over denominator 4 end fraction equals fraction numerator negative 12 straight b over denominator square root of 144 plus 2 straight b squared end root. straight space square root of 3 straight b squared end root end fraction  Perhatikan space bahwa space ruas space kiri space bernilai space positif comma space sehingga space pada  ruas space kanan space haruslah space straight b less than 0. space Selanjutnya comma  fraction numerator square root of 3 over denominator 4 end fraction equals fraction numerator negative 12 straight b over denominator straight b square root of 3 straight space square root of 144 plus 2 straight b squared end root end fraction  fraction numerator square root of 3 over denominator 4 end fraction equals fraction numerator negative 12 over denominator square root of 3 square root of 144 plus 2 straight b squared end root end fraction  3 straight space open parentheses square root of 144 plus 2 straight b squared end root close parentheses equals negative 12 straight space open parentheses 4 close parentheses  square root of 144 plus 2 straight b squared end root equals negative 16  144 plus 2 straight b squared equals 256  2 straight b squared equals 256 minus 144  straight b squared equals 112 over 2  straight b equals square root of 54  straight b equals plus-or-minus 2 square root of 14  Karena space straight b less than 0 comma space maka space straight b equals negative 2 square root of 14    bold Catatan bold space bold colon  Perlu space diperhatikan space bahwa space pada space pilihan space jawaban  tidak space ada space option space yang space bernilai space negatif.  Untuk space straight b equals 2 square root of 14 space sebenarnya space memberikan space nilai  cos invisible function application space straight theta equals negative fraction numerator square root of 3 over denominator 4 end fraction. space Namun space nilai space ini space mendekati space nilai  straight b space yang space sebenarnya space dicari. end style

Jawaban terverifikasi

Dijawab oleh:

F. Kurnia

Mahasiswa/Alumni Universitas Jember

Terakhir diupdate 04 Oktober 2021

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