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

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.

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

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

nilai cos theta dari vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell negative fraction numerator square root of 3 over denominator 6 end fraction end cell end table

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold minus fraction numerator square root of bold 3 over denominator bold 6 end fraction end cell end table.

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

  • 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 x subscript 1 i with hat on top plus y subscript 1 j with hat on top plus z subscript 1 k 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 plus z subscript 2 k with hat on top close parentheses end cell row blank equals cell x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 plus z subscript 1 z subscript 2 end cell end table  
  • 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 plus open parentheses z 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 negative i with hat on top minus j with hat on top dan b with rightwards arrow on top equals negative i with hat on top plus 2 j with hat on top plus k with hat on top  

Ditanya : Nilai cos theta dari 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 negative 1 times negative 1 right parenthesis plus left parenthesis negative 1 times 2 right parenthesis plus left parenthesis 0 times 1 right parenthesis over denominator open parentheses square root of left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared plus 0 squared end root close parentheses open parentheses square root of left parenthesis negative 1 right parenthesis squared plus 2 to the power of 2 end exponent plus 1 squared end root close parentheses end fraction end cell row blank equals cell fraction numerator 1 plus left parenthesis negative 2 right parenthesis plus 0 over denominator left parenthesis square root of 1 plus 1 plus 0 end root right parenthesis left parenthesis square root of 1 plus 4 plus 1 end root right parenthesis end fraction end cell row blank equals cell fraction numerator negative 1 over denominator square root of 2 times square root of 6 end fraction end cell row blank equals cell fraction numerator negative 1 over denominator square root of 12 end fraction cross times fraction numerator square root of 12 over denominator square root of 12 end fraction end cell row blank equals cell fraction numerator negative square root of 4 times 3 end root over denominator 12 end fraction end cell row blank equals cell fraction numerator negative 2 square root of 3 over denominator 12 end fraction end cell row blank equals cell negative fraction numerator square root of 3 over denominator 6 end fraction end cell end table 

Dengan demikian, nilai cos theta dari vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell negative fraction numerator square root of 3 over denominator 6 end fraction end cell end table

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