Diketahui vektor a=3i+5j​−2k, vektor b=−i−2j​+3k dan vektor c=2i−j​+4k. Tentukan proyeksi skalar ortogonal vektor b pada arah vektor (a−2c).

Pertanyaan

Diketahui vektor a with rightwards arrow on top equals 3 i with hat on top plus 5 j with hat on top minus 2 k with hat on top, vektor b with rightwards arrow on top equals negative i with hat on top minus 2 j with hat on top plus 3 k with hat on top dan vektor c with rightwards arrow on top equals 2 i with hat on top minus j with hat on top plus 4 k with hat on top.

  1. Tentukan proyeksi skalar ortogonal vektor b with rightwards arrow on top pada arah vektor open parentheses a with rightwards arrow on top minus 2 c with rightwards arrow on top close parentheses.

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Jawaban terverifikasi

Jawaban

 proyeksi skalar ortogonal vektor b with rightwards arrow on top pada arah vektor open parentheses a with rightwards arrow on top minus 2 c with rightwards arrow on top close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell negative fraction numerator 43 over denominator 5 square root of 6 end fraction end cell end table.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold minus fraction numerator bold 43 over denominator bold 5 square root of bold 6 end fraction end cell end table.

Ingat!

  • Jika diketahui vektor di bidang r with rightwards arrow on top equals x i with hat on top plus y j with hat on top maka dapat ditulis dalam bentuk vektor kolom sebagai berikut:

r with rightwards arrow on top equals open parentheses table row x row y end table close parentheses

  • Misalkan vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top adalah vektor-vektor sembarang, dan vektor c with rightwards arrow on top adalah proyeksi skalar ortogonal dari arah vektor a with rightwards arrow on top pada arah vektor b with rightwards arrow on top maka proyeksi skalar ortogonal dari vektor a with rightwards arrow on top pada arah vektor b with rightwards arrow on top ditentukan oleh:

c with rightwards arrow on top equals fraction numerator a with rightwards arrow on top space. space b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction

  • Rumus untuk menentukan panjang vektor r with rightwards arrow on top equals open parentheses table row x row y end table close parentheses adalah sebagai  berikut:

open vertical bar r with rightwards arrow on top close vertical bar equals square root of x squared plus y squared end root 

  • Rumus untuk menentukan hasil kali a with rightwards arrow on top space. space b with rightwards arrow on top jika diketahui vektor a with rightwards arrow on top equals open parentheses table row cell x subscript 1 end cell row cell y subscript 1 end cell end table close parentheses dan vektor b with rightwards arrow on top equals open parentheses table row cell x subscript 2 end cell row cell y subscript 2 end cell end table close parentheses adalah sebagai berikut:

a with rightwards arrow on top times b with rightwards arrow on top equals x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 

  • Rumus untuk perkalian skalar m dengan vektor a with rightwards arrow on top equals open parentheses table row cell x subscript 1 end cell row cell y subscript 1 end cell end table close parentheses adalah sebagai berikut:

m a with rightwards arrow on top equals m open parentheses table row cell x subscript 1 end cell row cell x subscript 2 end cell end table close parentheses equals open parentheses table row cell m x subscript 1 end cell row cell m x subscript 2 end cell end table close parentheses 

Diketahui:

Vektor a with rightwards arrow on top equals 3 i with hat on top plus 5 j with hat on top minus 2 k with hat on top rightwards arrow a with rightwards arrow on top equals open parentheses table row 3 row 5 row cell negative 2 end cell end table close parentheses 

Vektor b with rightwards arrow on top equals negative i with hat on top minus 2 j with hat on top plus 3 k with hat on top rightwards arrow b with rightwards arrow on top equals open parentheses table row cell negative 1 end cell row cell negative 2 end cell row 3 end table close parentheses 

Vektor c with rightwards arrow on top equals 2 i with hat on top minus j with hat on top plus 4 k with hat on top rightwards arrow open parentheses table row 2 row cell negative 1 end cell row 4 end table close parentheses 

Ditanya: 

Proyeksi skalar ortogonal vektor b with rightwards arrow on top pada arah vektor open parentheses a with rightwards arrow on top minus 2 c with rightwards arrow on top close parentheses.

Jawab:

Hasil dari vektor a with rightwards arrow on top minus 2 c with rightwards arrow on top adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top minus 2 c with rightwards arrow on top end cell equals cell open parentheses table row 3 row 5 row cell negative 2 end cell end table close parentheses minus 2 open parentheses table row 2 row cell negative 1 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses table row 3 row 5 row cell negative 2 end cell end table close parentheses minus open parentheses table row 4 row cell negative 2 end cell row 8 end table close parentheses end cell row blank equals cell open parentheses table row cell 3 minus 4 end cell row cell 5 plus 2 end cell row cell negative 2 minus 8 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell row 7 row cell negative 10 end cell end table close parentheses end cell end table

Jadi, proyeksi skalar ortogonal vektor b with rightwards arrow on top pada arah vektor open parentheses a with rightwards arrow on top minus 2 c with rightwards arrow on top close parentheses adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell p with rightwards arrow on top end cell equals cell fraction numerator b with rightwards arrow on top space. space open parentheses a with rightwards arrow on top minus 2 c with rightwards arrow on top close parentheses over denominator open vertical bar open parentheses a with rightwards arrow on top minus 2 c with rightwards arrow on top close parentheses close vertical bar end fraction end cell row blank equals cell fraction numerator open parentheses negative 1 close parentheses cross times open parentheses negative 1 close parentheses plus open parentheses negative 2 close parentheses cross times 7 plus 3 cross times open parentheses negative 10 close parentheses over denominator square root of open parentheses negative 1 close parentheses squared plus 7 squared plus open parentheses negative 10 close parentheses squared end root end fraction end cell row blank equals cell fraction numerator 1 minus 14 minus 30 over denominator square root of 1 plus 49 plus 100 end root end fraction end cell row blank equals cell fraction numerator negative 43 over denominator square root of 150 end fraction end cell row blank equals cell negative fraction numerator 43 over denominator square root of 25 cross times square root of 6 end fraction end cell row blank equals cell negative fraction numerator 43 over denominator 5 square root of 6 end fraction end cell end table

Dengan demikian, proyeksi skalar ortogonal vektor b with rightwards arrow on top pada arah vektor open parentheses a with rightwards arrow on top minus 2 c with rightwards arrow on top close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell negative fraction numerator 43 over denominator 5 square root of 6 end fraction end cell end table.

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Pertanyaan serupa

Diketahui vektor a=3i+5j​−2k, vektor b=−i−2j​+3k dan vektor c=2i−j​+4k. b. Tentukan proyeksi vektor ortogonal vektor b pada arah vektor (a−2c).

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Jawaban terverifikasi

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