Diketahui vektor-vektor p​=⎝⎛​−330​⎠⎞​ dan q​=⎝⎛​−242​⎠⎞​. Besar sudut antara p​ dan q​ adalah ....

Pertanyaan

Diketahui vektor-vektor straight p with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 3 row 0 end table close parentheses dan straight q with rightwards arrow on top equals open parentheses table row cell negative 2 end cell row 4 row 2 end table close parentheses. Besar sudut antara straight p with rightwards arrow on top dan straight q with rightwards arrow on top adalah ....

  1. 30 degree

  2. 45 degree

  3. 60 degree

  4. 90 degree

  5. 120 degree

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Jawaban yang benar dari pertanyaan tersebut adalah A.

Ingat rumus proyeksi skalar a with rightwards arrow on top pada b with rightwards arrow on top yaitu : 

open vertical bar straight a with rightwards arrow on top close vertical bar space cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction

Jika besar sudut antara straight p with rightwards arrow on top dan straight q with rightwards arrow on top adalah theta maka dapat dicari seperti berikut : 

space cos space straight theta space equals space fraction numerator p with rightwards arrow on top q with rightwards arrow on top over denominator open vertical bar p with rightwards arrow on top close vertical bar open vertical bar q with rightwards arrow on top close vertical bar end fraction

Perhatikan bahwa jika straight a with rightwards arrow on top equals open parentheses table row straight x row straight y row straight z end table close parentheses, maka open vertical bar straight a with rightwards arrow on top close vertical bar equals square root of straight x squared plus straight y squared plus straight z squared end root.  Besar straight p with rightwards arrow on top space dan space straight q with rightwards arrow on top dapat dicari terlebih dahulu yaitu : 

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight p with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 3 close parentheses squared plus 3 squared plus 0 squared end root end cell row blank equals cell square root of 9 plus 9 plus 0 end root end cell row blank equals cell square root of 18 end cell row cell open vertical bar straight q with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 2 close parentheses squared plus 4 squared plus 2 squared end root end cell row blank equals cell square root of 4 plus 16 plus 4 end root end cell row blank equals cell square root of 24 end cell row blank blank blank end table

substitusikan kedalam rumus maka menjadi seperti berikut : 

table attributes columnalign right center left columnspacing 0px end attributes row cell space cos space straight theta space end cell equals cell space fraction numerator open parentheses table row cell negative 3 end cell row 3 row 0 end table close parentheses times open parentheses table row cell negative 2 end cell row 4 row 2 end table close parentheses over denominator square root of 18 times square root of 24 end fraction end cell row cell space cos space straight theta end cell equals cell fraction numerator open parentheses negative 3 close parentheses open parentheses negative 2 close parentheses plus open parentheses 3 close parentheses open parentheses 4 close parentheses plus open parentheses 0 close parentheses open parentheses 2 close parentheses over denominator square root of 432 end fraction end cell row cell space cos space straight theta end cell equals cell fraction numerator 6 plus 12 plus 0 over denominator 12 square root of 3 end fraction end cell row cell space cos space straight theta end cell equals cell fraction numerator 18 over denominator 12 square root of 3 end fraction end cell row cell space cos space straight theta end cell equals cell fraction numerator 3 over denominator 2 square root of 3 end fraction. fraction numerator square root of 3 over denominator square root of 3 end fraction left parenthesis perkalian space sekawan right parenthesis end cell row cell space cos space straight theta end cell equals cell 3 over 2 times fraction numerator square root of 3 over denominator 3 end fraction end cell row cell space cos space straight theta end cell equals cell 1 half square root of 3 end cell row straight theta equals cell 30 degree end cell end table

Oleh karena itu, jawaban yang benar adalah A.

259

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