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Pertanyaan

Diketahui begin mathsize 14px style italic a with rightwards arrow on top equals open parentheses table row cell square root of 3 end cell row cell negative sign 2 end cell row 3 end table close parentheses end style, begin mathsize 14px style italic b with rightwards arrow on top equals open parentheses table row italic m row 2 row cell negative sign square root of 3 end cell end table close parentheses end style, dan begin mathsize 14px style c with rightwards arrow on top equals open parentheses table row italic n row 0 row cell 2 square root of 3 end cell end table close parentheses end style. Diketahui pula panjang proyeksi skalar undefinedpada begin mathsize 14px style a with rightwards arrow on top end style adalah 1 dan undefined tegak lurus dengan undefined. Jika m adalah bilangan bulat, maka nilai dari begin mathsize 14px style m over n end style  adalah ....

  1. begin mathsize 14px style 2 over 3 end style 

  2. begin mathsize 14px style 3 over 2 end style 

  3. begin mathsize 14px style 2 end style 

  4. begin mathsize 14px style 3 end style  

  5. begin mathsize 14px style 6 end style 

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah B.

Pembahasan

Misalkan proyeksi skalar undefined pada begin mathsize 14px style italic a with italic rightwards arrow on top end style adalah begin mathsize 14px style italic p with rightwards arrow on top end style. Karena panjang proyeksinya adalah 1, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar italic p with rightwards arrow on top close vertical bar end cell equals 1 row cell open vertical bar fraction numerator italic a with rightwards arrow on top ⋅ italic b with rightwards arrow on top over denominator open vertical bar italic a with rightwards arrow on top close vertical bar end fraction close vertical bar end cell equals 1 row cell open vertical bar fraction numerator open parentheses table row cell square root of 3 end cell row cell negative sign 2 end cell row 3 end table close parentheses ⋅ open parentheses table row italic m row 2 row cell negative sign square root of 3 end cell end table close parentheses over denominator square root of open parentheses square root of 3 close parentheses squared plus open parentheses negative sign 2 close parentheses squared plus 3 squared end root end fraction close vertical bar end cell equals 1 row cell open vertical bar fraction numerator square root of 3 ⋅ italic m plus open parentheses negative sign 2 close parentheses ⋅ 2 plus 3 ⋅ open parentheses negative sign square root of 3 close parentheses over denominator square root of 3 plus 4 plus 9 end root end fraction close vertical bar end cell equals 1 row cell open vertical bar fraction numerator italic m square root of 3 minus sign 4 minus sign 3 square root of 3 over denominator 4 end fraction close vertical bar end cell equals 1 end table 

Jadi, begin mathsize 14px style fraction numerator italic m square root of 3 minus sign 4 minus sign 3 square root of 3 over denominator 4 end fraction equals 1 space atau space fraction numerator italic m square root of 3 minus sign 4 minus sign 3 square root of 3 over denominator 4 end fraction equals minus sign 1 end style

Untuk begin mathsize 14px style fraction numerator italic m square root of 3 minus sign 4 minus sign 3 square root of 3 over denominator 4 end fraction equals 1 end style, maka kita dapatkan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator italic m square root of 3 minus sign 4 minus sign 3 square root of 3 over denominator 4 end fraction end cell equals 1 row cell italic m square root of 3 minus sign 4 minus sign 3 square root of 3 end cell equals 4 row cell italic m square root of 3 minus sign 3 square root of 3 end cell equals 8 row cell italic m square root of 3 end cell equals cell 8 plus 3 square root of 3 end cell row italic m equals cell fraction numerator 8 plus 3 square root of 3 over denominator square root of 3 end fraction end cell end table end style 

Perhatikan bahwa m bukanlah bilangan bulat, sehingga kasus ini tidaklah memenuhi kriteria yang diminta.

Selanjutnya untukbegin mathsize 14px style fraction numerator italic m square root of 3 minus sign 4 minus sign 3 square root of 3 over denominator 4 end fraction equals minus sign 1 end style,maka kita dapatkan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator italic m square root of 3 minus sign 4 minus sign 3 square root of 3 over denominator 4 end fraction end cell equals cell negative sign 1 end cell row cell italic m square root of 3 minus sign 4 minus sign 3 square root of 3 end cell equals cell negative sign 4 end cell row cell italic m square root of 3 minus sign 3 square root of 3 end cell equals 0 row cell italic m square root of 3 end cell equals cell 3 square root of 3 end cell row italic m equals cell fraction numerator 3 square root of 3 over denominator square root of 3 end fraction end cell row italic m equals 3 row blank blank blank end table end style 

Perhatikan bahwa m adalah bilangan bulat, sehingga kasus ini memenuhi kriteria yang diminta.
Jadi, didapat

begin mathsize 14px style italic b with rightwards arrow on top equals open parentheses table row italic m row 2 row cell negative sign square root of 3 end cell end table close parentheses equals open parentheses table row 3 row 2 row cell negative sign square root of 3 end cell end table close parentheses end style 

Karena begin mathsize 14px style italic b with rightwards arrow on top end style tegak lurus dengan begin mathsize 14px style italic c with rightwards arrow on top end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell italic b with rightwards arrow on top ⋅ italic c with rightwards arrow on top end cell equals 0 row cell open parentheses table row 3 row 2 row cell negative sign square root of 3 end cell end table close parentheses ⋅ open parentheses table row italic n row 0 row cell 2 square root of 3 end cell end table close parentheses end cell equals 0 row cell 3 ⋅ italic n plus 2 ⋅ 0 plus open parentheses negative sign square root of 3 close parentheses ⋅ 2 square root of 3 end cell equals 0 row cell 3 italic n plus 0 minus sign 6 end cell equals 0 row cell 3 italic n minus sign 6 end cell equals 0 row cell 3 italic n end cell equals 6 row italic n equals 2 end table end style 

Jadi,

begin mathsize 14px style italic m over italic n equals 3 over 2 end style 

Jadi, jawaban yang tepat adalah B.

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