Diketahui titik A(1, 2, 2), titik B(0, 1, 0), dan titik C(2,−1,−1). Tentukan ruas garis berarah AB dan ruas garis berarah AC dalam bentuk vektor kolom. Tentukan proyeksi vektor ortogonal AB pada arah AC dan proyeksi vektor ortogonal AC pada arah AB.

Pertanyaan

Diketahui titik straight A open parentheses 1 comma space 2 comma space 2 close parentheses, titik straight B open parentheses 0 comma space 1 comma space 0 close parentheses, dan titik straight C open parentheses 2 comma negative 1 comma negative 1 close parentheses.

  1. Tentukan ruas garis berarah AB with rightwards arrow on top dan ruas garis berarah AC with rightwards arrow on top dalam bentuk vektor kolom.
  2. Tentukan proyeksi vektor ortogonal AB with rightwards arrow on top pada arah AC with rightwards arrow on top dan proyeksi vektor ortogonal AC with rightwards arrow on top pada arah AB with rightwards arrow on top.

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

 

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah

  1.  table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell bold minus bold 1 end cell row cell bold minus bold 1 end cell row cell bold minus bold 2 end cell end table close parentheses end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell bold space bold space bold space bold 1 end cell row cell bold minus bold 3 end cell row cell bold minus bold 3 end cell end table close parentheses end cell end table
  2.  table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell bold 8 over bold 19 end cell row cell bold minus bold 24 over bold 19 end cell row cell bold minus bold 24 over bold 19 end cell end table close parentheses end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell bold minus bold 4 over bold 3 end cell row cell bold minus bold 4 over bold 3 end cell row cell bold minus bold 8 over bold 3 end cell end table close parentheses end cell end table

Ingat!

  • Jika koordinat titik straight A open parentheses x subscript 1 comma space y subscript 1 close parentheses dan straight B open parentheses x subscript 2 comma space y subscript 2 close parentheses maka dapat ditetapkan:

stack A B with rightwards arrow on top equals open parentheses table row cell x subscript 2 minus x subscript 1 end cell row cell y subscript 2 minus y subscript 1 end cell 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 vektor a with rightwards arrow on top pada arah vektor b with rightwards arrow on top maka proyeksi vektor 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 open parentheses 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 squared end fraction close parentheses b with rightwards arrow on top  

  • 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: 

Titik straight A open parentheses 1 comma space 2 comma space 2 close parentheses

Titik straight B open parentheses 0 comma space 1 comma space 0 close parentheses

Titik straight C open parentheses 2 comma negative 1 comma negative 1 close parentheses 

Ditanya:

  1. Ruas garis berarah stack A B with rightwards arrow on top dan ruas garis berarah stack A C with rightwards arrow on top dalam bentuk vektor kolom.
  2. Proyeksi vektor ortogonal stack A B with rightwards arrow on top pada arah stack A C with rightwards arrow on top dan proyeksi vektor ortogonal stack A C with rightwards arrow on top pada arah stack A B with rightwards arrow on top.

Jawab:

Ruas garis berarah stack A B with rightwards arrow on top adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell stack A B with rightwards arrow on top end cell equals cell open parentheses table row cell 0 minus 1 end cell row cell 1 minus 2 end cell row cell 0 minus 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell row cell negative 1 end cell row cell negative 2 end cell end table close parentheses end cell end table 

Ruas garis berarah stack A C with rightwards arrow on top adalah sebagai berikut:

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

Proyeksi vektor ortogonal stack A B with rightwards arrow on top pada arah stack A C with rightwards arrow on top adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell open parentheses fraction numerator stack A B with rightwards arrow on top space. space stack A C with rightwards arrow on top over denominator open vertical bar stack A C with rightwards arrow on top close vertical bar squared end fraction close parentheses stack A C with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator negative 1 cross times 1 plus open parentheses negative 1 close parentheses cross times open parentheses negative 3 close parentheses plus open parentheses negative 2 close parentheses cross times open parentheses negative 3 close parentheses over denominator open parentheses square root of 1 squared plus open parentheses negative 3 close parentheses squared plus open parentheses negative 3 close parentheses squared end root close parentheses squared end fraction close parentheses open parentheses table row cell space space 1 end cell row cell negative 3 end cell row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses fraction numerator negative 1 plus 3 plus 6 over denominator 1 plus 9 plus 9 end fraction close parentheses open parentheses table row 1 row cell negative 3 end cell row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses 8 over 19 close parentheses open parentheses table row 1 row cell negative 3 end cell row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 8 over 19 end cell row cell negative 24 over 19 end cell row cell negative 24 over 19 end cell end table close parentheses end cell end table

Proyeksi vektor ortogonal stack A C with rightwards arrow on top pada arah stack A B with rightwards arrow on top adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell open parentheses fraction numerator stack A C with rightwards arrow on top space. space stack A B with rightwards arrow on top over denominator open vertical bar stack A B with rightwards arrow on top close vertical bar squared end fraction close parentheses stack A B with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator 1 cross times open parentheses negative 1 close parentheses plus open parentheses negative 3 close parentheses cross times open parentheses negative 1 close parentheses plus open parentheses negative 3 close parentheses cross times open parentheses negative 2 close parentheses over denominator open parentheses square root of 1 squared plus open parentheses negative 1 close parentheses squared plus open parentheses negative 2 close parentheses squared end root close parentheses squared end fraction close parentheses open parentheses table row cell negative 1 end cell row cell negative 1 end cell row cell negative 2 end cell end table close parentheses end cell row blank equals cell open parentheses fraction numerator negative 1 plus 3 plus 6 over denominator 1 plus 1 plus 4 end fraction close parentheses open parentheses table row cell negative 1 end cell row cell negative 1 end cell row cell negative 2 end cell end table close parentheses end cell row blank equals cell open parentheses 8 over 6 close parentheses open parentheses table row cell negative 1 end cell row cell negative 1 end cell row cell negative 2 end cell end table close parentheses end cell row blank equals cell open parentheses 4 over 3 close parentheses open parentheses table row cell negative 1 end cell row cell negative 1 end cell row cell negative 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 4 over 3 end cell row cell negative 4 over 3 end cell row cell negative 8 over 3 end cell end table close parentheses end cell end table 

Dengan demikian, 

  1. Ruas garis berarah stack A B with rightwards arrow on top  adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell negative 1 end cell row cell negative 1 end cell row cell negative 2 end cell end table close parentheses end cell end table dan ruas garis berarah stack A C with rightwards arrow on top adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell space space space 1 end cell row cell negative 3 end cell row cell negative 3 end cell end table close parentheses end cell end table.
  2. Proyeksi vektor ortogonal stack A B with rightwards arrow on top pada arah stack A C with rightwards arrow on top adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell 8 over 19 end cell row cell negative 24 over 19 end cell row cell negative 24 over 19 end cell end table close parentheses end cell end table dan proyeksi vektor ortogonal stack A C with rightwards arrow on top pada arah stack A B with rightwards arrow on top adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell negative 4 over 3 end cell row cell negative 4 over 3 end cell row cell negative 8 over 3 end cell end table close parentheses end cell end table.

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