Jika θ adalah besar sudut antara vektor a dan vektor b. Hitunglah nilai cosθ dari pasangan-pasangan vektor a dan vektor b berikut. c. a=2i−5j​ dan b=−2i+5j​

Pertanyaan

Jika theta adalah besar sudut antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top. Hitunglah nilai cos theta dari pasangan-pasangan vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top berikut.

c. a with rightwards arrow on top equals 2 i with hat on top minus 5 j with hat on top dan b with rightwards arrow on top equals negative 2 i with hat on top plus 5 j with hat on top 

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

 nilai cos theta antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on topadalah negative 1

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold minus bold 1.

Ingat, jika diketahui vektor a with rightwards arrow on top equals x subscript 1 i with hat on top plus y subscript 1 j with hat on top dan vektor b with rightwards arrow on top equals x subscript 2 i with hat on top plus y subscript 2 j with hat on top, maka:

  • a with rightwards arrow on top times b with rightwards arrow on top equals open parentheses x subscript 1 i with hat on top plus y subscript 1 j with hat on top close parentheses open parentheses x subscript 2 i with hat on top plus y subscript 2 j with hat on top close parentheses equals x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 
  • open vertical bar a with rightwards arrow on top close vertical bar equals square root of open parentheses x subscript 1 close parentheses squared plus open parentheses y subscript 1 close parentheses squared end root 
  • cos space theta equals 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 

Diketahui : a with rightwards arrow on top equals 2 i with hat on top minus 5 j with hat on top dan b with rightwards arrow on top equals negative 2 i with hat on top plus 5 j with hat on top 

Ditanya : nilai cos theta antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top.

Penyelesaian : 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space theta 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 left parenthesis 2 times negative 2 right parenthesis plus left parenthesis negative 5 times 5 right parenthesis over denominator open parentheses square root of 2 squared plus open parentheses negative 5 close parentheses squared end root close parentheses open parentheses square root of open parentheses negative 2 close parentheses squared plus 5 squared end root close parentheses end fraction end cell row blank equals cell fraction numerator negative 4 plus left parenthesis negative 25 right parenthesis over denominator left parenthesis square root of 4 plus 25 end root right parenthesis left parenthesis square root of 4 plus 25 end root right parenthesis end fraction end cell row blank equals cell fraction numerator negative 29 over denominator left parenthesis square root of 29 right parenthesis left parenthesis square root of 29 right parenthesis end fraction end cell row blank equals cell fraction numerator negative 29 over denominator 29 end fraction end cell row blank equals cell negative 1 end cell end table 

Dengan demikian, nilai cos theta antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on topadalah negative 1

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