Diketahui ∣∣​a∣∣​=3, ∣∣​b∣∣​=5, dan ∣∣​a−b∣∣​=211​. Nilai ∣∣​a+b∣∣​ adalah ....

Pertanyaan

Diketahui open vertical bar a with rightwards arrow on top close vertical bar equals 3open vertical bar b with rightwards arrow on top close vertical bar equals 5, dan open vertical bar a with rightwards arrow on top minus b with rightwards arrow on top close vertical bar equals 2 square root of 11. Nilai open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar adalah ....

  1. 3 square root of 2

  2. 2 square root of 5

  3. 2 square root of 6

  4. 3 square root of 3

  5. 4 square root of 2

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah C.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah C.

Ingat!

  • 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 penjumlahan dan pengurangan dua vektor, jika diketahui 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 maka

a with rightwards arrow on top plus-or-minus b 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 plus-or-minus open parentheses table row cell x subscript 2 end cell row cell y subscript 2 end cell end table close parentheses equals open parentheses table row cell x subscript 1 plus-or-minus x subscript 2 end cell row cell y subscript 1 plus-or-minus y subscript 2 end cell end table close parentheses 

Diketahui:  open vertical bar a with rightwards arrow on top close vertical bar equals 3open vertical bar b with rightwards arrow on top close vertical bar equals 5, dan open vertical bar a with rightwards arrow on top minus b with rightwards arrow on top close vertical bar equals 2 square root of 11.

Ditanya: nilai open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar.

Jawab:

Misalkan 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, dengan menggunakan rumus-rumus di atas, maka diperoleh hubungan sebagai berikut:

  • Untuk vektor a with rightwards arrow on top 

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell square root of x subscript 1 squared plus y subscript 1 squared end root end cell row 3 equals cell square root of x subscript 1 squared plus y subscript 1 squared end root end cell row cell 3 squared end cell equals cell x subscript 1 squared plus y subscript 1 squared end cell row 9 equals cell x subscript 1 squared plus y subscript 1 squared end cell row blank left right double arrow cell x subscript 1 squared plus y subscript 1 squared equals 9 end cell end table 

  • Untuk vektor b with rightwards arrow on top

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar b with rightwards arrow on top close vertical bar end cell equals cell square root of x subscript 2 squared plus y subscript 2 squared end root end cell row 5 equals cell square root of x subscript 2 squared plus y subscript 2 squared end root end cell row cell 5 squared end cell equals cell x subscript 2 squared plus y subscript 2 squared end cell row 25 equals cell x subscript 2 squared plus y subscript 2 squared end cell row blank left right double arrow cell x subscript 2 squared plus y subscript 2 squared equals 25 end cell end table 

  • Untuk vektor a with rightwards arrow on top minus b with rightwards arrow on top 

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a with rightwards arrow on top minus b with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses x subscript 1 minus x subscript 2 close parentheses squared plus open parentheses y subscript 1 minus y subscript 2 close parentheses squared end root end cell row cell 2 square root of 11 end cell equals cell square root of open parentheses x subscript 1 squared minus 2 x subscript 1 x subscript 2 plus x subscript 2 squared close parentheses plus open parentheses y subscript 1 squared minus 2 y subscript 1 y subscript 2 plus y subscript 2 squared close parentheses end root end cell row cell 2 square root of 11 end cell equals cell square root of x subscript 1 squared plus y subscript 1 squared minus 2 x subscript 1 x subscript 2 minus 2 y subscript 1 y subscript 2 plus x subscript 2 squared plus y subscript 2 squared end root end cell row cell 2 square root of 11 end cell equals cell square root of open parentheses x subscript 1 squared plus y subscript 1 squared close parentheses minus 2 open parentheses x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 close parentheses plus open parentheses x subscript 2 squared plus y subscript 2 squared close parentheses end root end cell row cell 2 square root of 11 end cell equals cell square root of 9 minus 2 open parentheses x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 close parentheses plus 25 end root end cell row cell 2 square root of 11 end cell equals cell square root of 34 minus 2 open parentheses x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 close parentheses end root end cell row cell open parentheses 2 square root of 11 close parentheses squared end cell equals cell 34 minus 2 open parentheses x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 close parentheses end cell row 44 equals cell 34 minus 2 open parentheses x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 close parentheses end cell row cell 44 minus 34 end cell equals cell negative 2 open parentheses x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 close parentheses end cell row blank left right double arrow cell x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 equals fraction numerator 10 over denominator negative 2 end fraction end cell row cell x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 end cell equals cell negative 5 end cell end table  

Dengan demikian, maka nilai open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses x subscript 1 plus x subscript 2 close parentheses squared plus open parentheses y subscript 1 plus y subscript 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses x subscript 1 squared plus 2 x subscript 1 x subscript 2 plus x subscript 2 squared close parentheses plus open parentheses y subscript 1 squared plus 2 y subscript 1 y subscript 2 plus y subscript 2 squared close parentheses end root end cell row blank equals cell square root of x subscript 1 squared plus y subscript 1 squared plus 2 x subscript 1 x subscript 2 plus 2 y subscript 1 y subscript 2 plus x subscript 2 squared plus y subscript 2 squared end root end cell row blank equals cell square root of open parentheses x subscript 1 squared plus y subscript 1 squared close parentheses plus 2 open parentheses x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 close parentheses plus open parentheses x subscript 2 squared plus y subscript 2 squared close parentheses end root end cell row blank equals cell square root of 9 plus 2 open parentheses negative 5 close parentheses plus 25 end root end cell row blank equals cell square root of 9 minus 10 plus 25 end root end cell row blank equals cell square root of 24 end cell row blank equals cell square root of 4 cross times 6 end root end cell row blank equals cell 2 square root of 6 end cell end table 

Oleh karena itu, jawaban yang benar adalah C.

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Jason Andriy Putra Pratama Muga

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Airine Caroline

Pembahasan lengkap banget

Pertanyaan serupa

Misalkan diketahui vektor a=⎝⎛​1−23​−213​​⎠⎞​ dan vektor b=⎝⎛​−123​21​​⎠⎞​.  a. tentukan vektor (a+b) dan vektor (a−b).  b. Jika θ adalah sudut antara vektor (a+b) dan vektor (a−b), hitunglah nilai ...

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