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Tentukan bayangan: a. garis≡x3​−y=0 jika dirotasi oleh [(1,2),θ=30∘]

Pertanyaan

Tentukan bayangan:

a. begin mathsize 14px style garis identical to x square root of 3 minus y equals 0 end style jika dirotasi oleh open square brackets open parentheses 1 comma 2 close parentheses comma space theta equals 30 degree close square brackets 

Pembahasan Soal:

Menentukan bayangan garis jika dirotasi oleh open square brackets open parentheses 1 comma 2 close parentheses comma space theta equals 30 degree close square brackets:

Kita akan melakukan dengan cara eliminasi Gauss-Jordan dalam menjawab soal ini.

Persamaan matriks Gauss-Jordan:

open parentheses table row cell cos space 30 degree end cell cell negative sin space 30 degree end cell row cell sin space 30 degree end cell cell cos space 30 degree end cell end table vertical ellipsis table row cell x apostrophe minus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses equals open parentheses table row cell 1 half square root of 3 end cell cell negative 1 half end cell row cell 1 half end cell cell 1 half square root of 3 end cell end table vertical ellipsis table row cell x apostrophe minus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses

Proses Eliminasi Gauss-Jordan:

left right double arrow open parentheses table row cell 1 half square root of 3 end cell cell negative 1 half end cell row cell 1 half end cell cell 1 half square root of 3 end cell end table vertical ellipsis table row cell x apostrophe minus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses table row cell fraction numerator 2 over denominator square root of 3 end fraction B subscript 2 end cell row cell fraction numerator 2 over denominator square root of 3 end fraction B subscript 2 end cell end table left right double arrow open parentheses table row 1 cell negative fraction numerator square root of 3 over denominator 3 end fraction end cell row cell fraction numerator square root of 3 over denominator 3 end fraction end cell 1 end table vertical ellipsis table row cell fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe minus fraction numerator 2 square root of 3 over denominator 3 end fraction end cell row cell fraction numerator 2 square root of 3 over denominator 3 end fraction y apostrophe minus fraction numerator 4 square root of 3 over denominator 3 end fraction end cell end table close parentheses table row blank row cell negative fraction numerator square root of 3 over denominator 3 end fraction B subscript 1 plus B subscript 2 end cell end table left right double arrow open parentheses table row 1 cell negative fraction numerator square root of 3 over denominator 3 end fraction end cell row 0 cell 4 over 3 end cell end table vertical ellipsis table row cell fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe minus fraction numerator 2 square root of 3 over denominator 3 end fraction end cell row cell negative 2 over 3 x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction y apostrophe plus 2 over 3 minus 4 over 3 square root of 3 end cell end table close parentheses table row blank row cell 3 over 4 B subscript 2 end cell end table left right double arrow open parentheses table row 1 cell negative fraction numerator square root of 3 over denominator 3 end fraction end cell row 0 1 end table vertical ellipsis table row cell fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe minus fraction numerator 8 square root of 3 over denominator 3 end fraction end cell row cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 1 half minus square root of 3 end cell end table close parentheses table row cell fraction numerator square root of 3 over denominator 3 end fraction B subscript 2 plus B subscript 1 end cell row blank end table left right double arrow open parentheses table row 1 0 row 0 1 end table vertical ellipsis table row cell negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction minus 1 end cell row cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 1 half minus square root of 3 end cell end table close parentheses

Hal ini berarti:

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 1 end cell equals cell negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction minus 1 end cell row x equals cell negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction space... left parenthesis 1 right parenthesis end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus 2 end cell equals cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 1 half minus square root of 3 end cell row y equals cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 5 over 2 minus square root of 3... left parenthesis 2 right parenthesis end cell end table

Kedua persamaan di atas disubstitusikan ke persamaan garis x square root of 3 minus y equals 0

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell x square root of 3 minus y end cell equals 0 row cell open parentheses negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction close parentheses square root of 3 minus open parentheses negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 5 over 2 minus square root of 3 close parentheses end cell equals 0 row cell open parentheses negative 1 half x apostrophe plus 2 x apostrophe plus fraction numerator square root of 3 over denominator 2 end fraction y apostrophe minus 1 half minus 8 close parentheses minus open parentheses negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 5 over 2 minus square root of 3 close parentheses end cell equals 0 row cell 2 x apostrophe minus 3 minus 8 minus square root of 3 end cell equals 0 row cell 2 x apostrophe minus 11 minus square root of 3 end cell equals 0 end table end style

Jadi, persamaan bayangan garis tersebut adalah 2 x apostrophe minus 11 minus square root of 3 equals 0.

 

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

O. Rahmawati

Mahasiswa/Alumni UIN Sunan Gunung Djati Bandung

Terakhir diupdate 06 Oktober 2021

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