Pertanyaan

Diketahui vektor a = − 4 i + 4 k dan vektor b = 2 i + 8 k − 2 k . Kosinus sudut antara vektor a dan vektor b adalah ...

Diketahui vektor  dan vektor . Kosinus sudut antara vektor  dan vektor  adalah ...

  1. 1 half 

  2. 1 third 

  3. 1 fourth 

  4. negative 1 third 

  5. negative 1 half 

H. Eka

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

jawaban yang tepat adalah D.

Pembahasan

Panjang vektor , yaitu Jika vektor dan , maka Jika vektor dan membentuk sudut , maka Asumsi soal: vektor Berdasarkan konsep di atas, dapat ditentukan panjang vektor dan sebagai berikut. Hasil operasi perkalian kedua vektor, yaitu Kosinus sudut antara vektor dan vektor dapat ditentukan sebagai berikut. Oleh karena itu, jawaban yang tepat adalah D.

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 dan b with rightwards arrow on top membentuk sudut alpha, maka 

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

Asumsi soal: vektor b with rightwards arrow on top equals 2 i with rightwards arrow on top plus 8 j with rightwards arrow on top minus 2 k with rightwards arrow on top

Berdasarkan konsep di atas, dapat ditentukan panjang vektor a with rightwards arrow on top dan b with rightwards arrow on top sebagai berikut.

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 open parentheses negative 4 close parentheses squared plus 0 squared plus 4 squared end root end cell row blank equals cell square root of 16 plus 16 end root end cell row blank equals cell square root of 32 end cell row blank equals cell 4 square root of 2 end cell end table

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 2 squared plus 8 squared plus open parentheses negative 2 close parentheses squared end root end cell row blank equals cell square root of 4 plus 64 plus open parentheses negative 2 close parentheses squared end root end cell row blank equals cell square root of 72 end cell row blank equals cell 6 square root of 2 end cell end table

Hasil operasi perkalian kedua vektor, yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell open parentheses table row cell negative 4 end cell row 0 row 4 end table close parentheses open parentheses table row 2 row 8 row cell negative 2 end cell end table close parentheses end cell row blank equals cell negative 4 times 2 plus 0 times 8 plus 4 times open parentheses negative 2 close parentheses end cell row blank equals cell negative 8 plus 0 minus 8 end cell row blank equals cell negative 16 end cell end table

Kosinus sudut antara vektor stack text a end text with rightwards arrow on top dan vektor stack text b end text with rightwards arrow on top dapat ditentukan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals cell 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 end cell 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 cell cos space alpha end cell equals cell fraction numerator negative 16 over denominator 4 square root of 2 times 6 square root of 2 end fraction end cell row cell cos space alpha end cell equals cell fraction numerator negative 16 over denominator 48 end fraction end cell row cell cos space alpha end cell equals cell negative 1 third end cell end table

Oleh karena itu, jawaban yang tepat adalah D.

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