DIketahui a=i−5j​+2k dan b=8i+xj​+6k. Jika panjang proyeksi skalar a pada arah vektor b adalah 0,2 ∣∣​b∣∣​, tentukan proyeksi vektor a yang searah b.  Petunjuk: terdapat dua jawaban

Pertanyaan

DIketahui a with rightwards arrow on top equals i with rightwards arrow on top minus 5 j with rightwards arrow on top plus 2 k with rightwards arrow on top dan b with rightwards arrow on top equals 8 i with rightwards arrow on top plus x j with rightwards arrow on top plus 6 k with rightwards arrow on top. Jika panjang proyeksi skalar a with rightwards arrow on top pada arah vektor b with rightwards arrow on top adalah 0 comma 2 space open vertical bar b with rightwards arrow on top close vertical bar, tentukan proyeksi vektor a with rightwards arrow on top yang searah b with rightwards arrow on top

Petunjuk: terdapat dua jawaban

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold 1 bold. bold 6 bold i with bold rightwards arrow on top bold plus bold 1 bold. bold 2 bold k with bold rightwards arrow on top end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold 1 bold. bold 6 bold i with bold rightwards arrow on top bold minus bold 3 bold j with bold rightwards arrow on top bold plus bold 1 bold. bold 2 bold k with bold rightwards arrow on top end cell end table

Ingat!

x with rightwards arrow on top equals x subscript 1 i with rightwards arrow on top plus x subscript 2 j with rightwards arrow on top plus x subscript 3 k with rightwards arrow on top y with rightwards arrow on top equals y subscript 1 i with rightwards arrow on top plus y subscript 2 j with rightwards arrow on top plus y subscript 3 k with rightwards arrow on top

maka 

1. Proyeksi space Skalar equals fraction numerator straight x with rightwards arrow on top times straight y with rightwards arrow on top over denominator open vertical bar straight y with rightwards arrow on top close vertical bar end fraction 

2. Proyeksi space Vektor equals Proyeksi space Skalar times fraction numerator straight y with rightwards arrow on top over denominator open vertical bar straight y with rightwards arrow on top close vertical bar end fraction 

dengan 

x with rightwards arrow on top times y with rightwards arrow on top equals x subscript 1 y subscript 1 plus x subscript 2 y subscript 2 plus x subscript 3 y subscript 3 open vertical bar y with rightwards arrow on top close vertical bar equals square root of y subscript 1 squared plus y subscript 2 squared plus y subscript 3 squared end root 

menggunakan definisi yang diberikan bahwa,

  • table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell equals cell 0.2 open vertical bar b with rightwards arrow on top close vertical bar end cell row cell 8 minus 5 x plus 12 end cell equals cell 0.2 open vertical bar b with rightwards arrow on top close vertical bar squared end cell row cell negative 5 x plus 20 end cell equals cell 1 fifth open parentheses square root of 64 plus x squared plus 36 end root close parentheses squared end cell row cell negative 25 x plus 100 end cell equals cell 100 plus x squared end cell row 0 equals cell x squared plus 25 x end cell row 0 equals cell x open parentheses x plus 25 close parentheses begin bold style left parenthesis difaktorkan right parenthesis end style end cell row x equals cell 0 logical or x equals negative 25 end cell end table 
  • table attributes columnalign right center left columnspacing 0px end attributes row cell Proyeksi space Vektor end cell equals cell 0.2 open vertical bar straight b with rightwards arrow on top close vertical bar times fraction numerator straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction begin bold style left parenthesis habis space membagi right parenthesis end style end cell row blank equals cell 0.2 straight b with rightwards arrow on top end cell end table 

Substitusikan nilai x equals 0 dan x equals negative 25 ke vektor b with rightwards arrow on top equals 8 i with rightwards arrow on top plus x j with rightwards arrow on top plus 6 k with rightwards arrow on top

  1. table attributes columnalign right center left columnspacing 0px end attributes row x equals cell 0 rightwards arrow b with rightwards arrow on top equals 8 i with rightwards arrow on top plus 6 k with rightwards arrow on top end cell row cell Proyksi space Vektor end cell equals cell 0.2 open parentheses 8 i with rightwards arrow on top plus 6 k with rightwards arrow on top close parentheses end cell row blank equals cell 1.6 i with rightwards arrow on top plus 1.2 k with rightwards arrow on top end cell end table 
  2. table attributes columnalign right center left columnspacing 0px end attributes row x equals cell negative 25 rightwards arrow b with rightwards arrow on top equals 8 i with rightwards arrow on top minus 25 j with rightwards arrow on top plus 6 k with rightwards arrow on top end cell row cell Proyksi space Vektor end cell equals cell 0.2 open parentheses 8 i with rightwards arrow on top minus 25 j with rightwards arrow on top plus 6 k with rightwards arrow on top close parentheses end cell row blank equals cell 1.6 i with rightwards arrow on top minus 3 j with rightwards arrow on top plus 1.2 k with rightwards arrow on top end cell end table 

dengan demikian, proyeksi vektor a with rightwards arrow on top yang searah b with rightwards arrow on topadalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1.6 i with rightwards arrow on top plus 1.2 k with rightwards arrow on top end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1.6 i with rightwards arrow on top minus 3 j with rightwards arrow on top plus 1.2 k with rightwards arrow on top end cell end table

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