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Diketahui vektor a=⎝⎛​x12​⎠⎞​ dan b=⎝⎛​23−1​⎠⎞​ dan panjang proyeksi vektor a pada vektor b adalah 71​14​. Sudut antara  dan  adalah α, maka cos α= ...

Pertanyaan

Diketahui vektor stack text a end text with rightwards arrow on top equals open parentheses table row x row 1 row 2 end table close parentheses dan stack text b end text with rightwards arrow on top equals open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses dan panjang proyeksi vektor stack text a end text with rightwards arrow on top pada vektor stack text b end text with rightwards arrow on top adalah 1 over 7 square root of 14. Sudut antara stack text a end text with rightwards arrow on top dan stack text b end text with rightwards arrow on top adalah alpha, maka cos space alpha equals space... 

  1. 1 over 7 square root of 7 

  2. 1 over 14 square root of 14 

  3. 1 over 21 square root of 21 

  4. 1 over 27 square root of 27 

  5. 1 over 28 square root of 28 

H. Eka

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

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Pembahasan

Panjang vektor a with rightwards arrow on top equals x i with rightwards arrow on top plus y j with rightwards arrow on top plus z k with rightwards arrow on top, yaitu open vertical bar a with rightwards arrow on top close vertical bar equals square root of x squared plus y squared plus z squared end root

Jika vektor a with rightwards arrow on top equals a subscript 1 i with rightwards arrow on top plus a subscript 2 j with rightwards arrow on top plus a subscript 3 k with rightwards arrow on top dan b with rightwards arrow on top equals b subscript 1 i with rightwards arrow on top plus b subscript 2 j with rightwards arrow on top plus b subscript 3 k with rightwards arrow on top, maka

a with rightwards arrow on top times b with rightwards arrow on top equals a subscript 1 times b subscript 1 plus a subscript 2 times b subscript 2 plus a subscript 3 times b subscript 3

Jika vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top membentuk sudut alpha, maka berlaku:

a with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha

Panjang proyeksi vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top dapat ditentukan menggunakan rumus berikut.

open vertical bar c with rightwards arrow on top close vertical bar equals fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction

Berdasarkan konsep di atas, nilai x dapat ditentukan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar c with rightwards arrow on top close vertical bar end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator open parentheses table row x row 1 row 2 end table close parentheses open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses over denominator square root of 2 squared plus 3 squared plus open parentheses negative 1 close parentheses squared end root end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator 2 x plus 1 times 3 plus 2 open parentheses negative 1 close parentheses over denominator square root of 4 plus 9 plus 1 end root end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator 2 x plus 1 over denominator square root of 14 end fraction end cell row cell 1 over 7 square root of 14 times square root of 14 end cell equals cell 2 x plus 1 end cell row cell 14 over 7 end cell equals cell 2 x plus 1 end cell row 2 equals cell 2 x plus 1 end cell row cell 2 x end cell equals 1 row x equals cell 1 half end cell end table

Diperoleh x equals 1 half sehingga a with rightwards arrow on top equals open parentheses table row cell 1 half end cell row 1 row 2 end table close parentheses

Nilai cos space alpha adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator open vertical bar a with rightwards arrow on top close vertical bar end fraction times fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of open parentheses begin display style 1 half end style close parentheses squared plus 1 squared plus 2 squared end root end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 1 over denominator square root of begin display style 21 over 4 end style end root end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 1 over denominator begin display style 1 half end style square root of 21 end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 2 over denominator square root of 21 end fraction times 1 over 7 square root of 14 end cell row blank equals cell 2 over 21 square root of 6 end cell end table

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