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Pertanyaan

Jika begin mathsize 14px style open parentheses u with rightwards arrow on top plus v with rightwards arrow on top close parentheses space d a n space open parentheses u with rightwards arrow on top minus v with rightwards arrow on top close parentheses end style saling tegak lurus dan begin mathsize 14px style open vertical bar u with rightwards arrow on top close vertical bar plus open vertical bar v with rightwards arrow on top close vertical bar equals 2 open vertical bar u with rightwards arrow on top minus v with rightwards arrow on top close vertical bar end style, maka besar sudut yang dibentuk oleh vektor begin mathsize 14px style u with rightwards arrow on top space d a n space v with rightwards arrow on top end style adalah ....

  1. 30°

  2. 45°

  3. 60°

  4. 120°

  5. 150°

A. Rizky

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Karena begin mathsize 14px style open parentheses u with rightwards arrow on top plus v with rightwards arrow on top close parentheses space d a n space open parentheses u with rightwards arrow on top minus v with rightwards arrow on top close parentheses end style saling tegak lurus, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses u with rightwards arrow on top plus v with rightwards arrow on top close parentheses times open parentheses u with rightwards arrow on top minus v with rightwards arrow on top close parentheses end cell equals 0 row cell u with rightwards arrow on top times u with rightwards arrow on top minus u with rightwards arrow on top times v with rightwards arrow on top plus v with rightwards arrow on top times u with rightwards arrow on top minus v with rightwards arrow on top times v with rightwards arrow on top end cell equals 0 row cell open vertical bar u with rightwards arrow on top close vertical bar open vertical bar u with rightwards arrow on top close vertical bar minus u with rightwards arrow on top times v with rightwards arrow on top plus u with rightwards arrow on top times v with rightwards arrow on top minus open vertical bar v with rightwards arrow on top close vertical bar open vertical bar v with rightwards arrow on top close vertical bar end cell equals 0 row cell open vertical bar u with rightwards arrow on top close vertical bar open vertical bar u with rightwards arrow on top close vertical bar minus open vertical bar v with rightwards arrow on top close vertical bar open vertical bar v with rightwards arrow on top close vertical bar end cell equals 0 row cell open vertical bar u with rightwards arrow on top close vertical bar squared minus open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals 0 row cell open vertical bar u with rightwards arrow on top close vertical bar squared end cell equals cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell row cell open vertical bar u with rightwards arrow on top close vertical bar end cell equals cell plus-or-minus open vertical bar v with rightwards arrow on top close vertical bar end cell end table end style

Karena panjang vektor  begin mathsize 14px style u with rightwards arrow on top end style tidak mungkin bernilai negatif, maka begin mathsize 14px style open vertical bar u with rightwards arrow on top close vertical bar equals open vertical bar v with rightwards arrow on top close vertical bar end style

Kemudian, perhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar u with rightwards arrow on top close vertical bar plus open vertical bar v with rightwards arrow on top close vertical bar end cell equals cell 2 open vertical bar u with rightwards arrow on top minus v with rightwards arrow on top close vertical bar end cell row cell open vertical bar v with rightwards arrow on top close vertical bar plus open vertical bar v with rightwards arrow on top close vertical bar end cell equals cell 2 open vertical bar u with rightwards arrow on top minus v with rightwards arrow on top close vertical bar end cell row cell 2 open vertical bar v with rightwards arrow on top close vertical bar end cell equals cell 2 open vertical bar u with rightwards arrow on top minus v with rightwards arrow on top close vertical bar end cell row cell open vertical bar v with rightwards arrow on top close vertical bar end cell equals cell open vertical bar u with rightwards arrow on top minus v with rightwards arrow on top close vertical bar end cell row cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals cell open vertical bar u with rightwards arrow on top minus v with rightwards arrow on top close vertical bar squared end cell row cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals cell open parentheses u with rightwards arrow on top minus v with rightwards arrow on top close parentheses times open parentheses u with rightwards arrow on top minus v with rightwards arrow on top close parentheses end cell row cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals cell u with rightwards arrow on top times u with rightwards arrow on top minus u with rightwards arrow on top times v with rightwards arrow on top minus v with rightwards arrow on top times u with rightwards arrow on top plus v with rightwards arrow on top times v with rightwards arrow on top end cell row cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals cell open vertical bar u with rightwards arrow on top close vertical bar squared minus u with rightwards arrow on top times v with rightwards arrow on top minus u with rightwards arrow on top times v with rightwards arrow on top plus open vertical bar v with rightwards arrow on top close vertical bar squared end cell row cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals cell open vertical bar u with rightwards arrow on top close vertical bar squared minus 2 u with rightwards arrow on top times v with rightwards arrow on top plus open vertical bar v with rightwards arrow on top close vertical bar squared end cell row cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals cell open vertical bar v with rightwards arrow on top close vertical bar squared minus 2 u with rightwards arrow on top times v with rightwards arrow on top plus open vertical bar v with rightwards arrow on top close vertical bar squared end cell row cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell equals cell 2 open vertical bar v with rightwards arrow on top close vertical bar squared minus 2 u with rightwards arrow on top times v with rightwards arrow on top end cell row cell 2 u with rightwards arrow on top times v with rightwards arrow on top end cell equals cell open vertical bar v with rightwards arrow on top close vertical bar squared end cell row cell u with rightwards arrow on top times v with rightwards arrow on top end cell equals cell 1 half open vertical bar v with rightwards arrow on top close vertical bar squared end cell end table end style 

Misalkan besar sudut yang dibentuk oleh vektor begin mathsize 14px style u with rightwards arrow on top space d a n space v with rightwards arrow on top end style adalah α. Oleh karena itu, kita dapatkan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar u with rightwards arrow on top close vertical bar open vertical bar v with rightwards arrow on top close vertical bar end fraction end cell row cell cos space alpha end cell equals cell fraction numerator 1 half open vertical bar v with rightwards arrow on top close vertical bar squared over denominator open vertical bar v with rightwards arrow on top close vertical bar open vertical bar v with rightwards arrow on top close vertical bar end fraction end cell row cell cos space alpha end cell equals cell fraction numerator 1 half open vertical bar v with rightwards arrow on top close vertical bar squared over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction end cell row cell cos space alpha end cell equals cell 1 half end cell row alpha equals cell 60 degree end cell row blank blank blank end table end style

Maka besar sudut yang dibentuk oleh vektor begin mathsize 14px style u with rightwards arrow on top space d a n space v with rightwards arrow on top end style adalah 60°.

Jadi, jawaban yang tepat adalah C.

 

 

 

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