Tentukan besar sudut yang dibentuk oleh dua vektor berikut:       b.  Vektor u=5i+j​−3k dan vektor v=2i+3j​+4k.

Pertanyaan

Tentukan besar sudut yang dibentuk oleh dua vektor berikut:

      b.  Vektor u with rightwards arrow on top equals 5 i with hat on top plus j with hat on top minus 3 k with hat on top dan vektor v with rightwards arrow on top equals 2 i with hat on top plus 3 j with hat on top plus 4 k with hat on top.

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Master Teacher

Jawaban terverifikasi

Jawaban

 besar sudut yang dibentuk oleh vektor u with rightwards arrow on top equals 5 i with hat on top plus j with hat on top minus 3 k with hat on top dan vektor v with rightwards arrow on top equals 2 i with hat on top plus 3 j with hat on top plus 4 k with hat on top adalah 88 comma 2 degree.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold 88 bold comma bold 2 bold degree

 Ingat!

  • Rumus untuk menentukan nilai cosinus antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top adalah sebagai berikut:

cos space theta equals fraction numerator a with rightwards arrow on top space. space 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 

  • Rumus untuk menentukan panjang vektor r with rightwards arrow on top equals x i with hat on top plus y j with hat on top plus z k with hat on top adalah sebagai  berikut:

open vertical bar r with rightwards arrow on top close vertical bar equals square root of x squared plus y squared plus z squared end root 

  • Rumus untuk menentukan hasil kali a with rightwards arrow on top space. space b with rightwards arrow on top 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 plus z subscript 1 k 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 plus z subscript 2 k with hat on top adalah sebagai berikut:

a with rightwards arrow on top. space b with rightwards arrow on top equals x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 plus z subscript 1 z subscript 2  

Diketahui:

Vektor u with rightwards arrow on top equals 5 i with hat on top plus j with hat on top minus 3 k with hat on top 

Vektor v with rightwards arrow on top equals 2 i with hat on top plus 3 j with hat on top plus 4 k with hat on top

Ditanya: besar sudut yang dibentuk dua vektor.

Jawab:

Dengan menggunakan rumus-rumus di atas, maka besar sudut yang dibentuk oleh vektor u with rightwards arrow on top dan vektor v with rightwards arrow on top adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space theta end cell equals cell fraction numerator u with rightwards arrow on top space. space v with rightwards arrow on top over denominator open vertical bar u with rightwards arrow on top close vertical bar open vertical bar v with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 5 cross times 2 plus 1 cross times 3 plus open parentheses negative 3 close parentheses cross times 4 over denominator open parentheses square root of 5 squared plus 1 squared plus open parentheses negative 3 close parentheses squared end root close parentheses open parentheses square root of 2 squared plus 3 squared plus 4 squared end root close parentheses end fraction end cell row blank equals cell fraction numerator 10 plus 3 minus 12 over denominator open parentheses square root of 25 plus 1 plus 9 end root close parentheses open parentheses square root of 4 plus 9 plus 16 end root close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator open parentheses square root of 35 close parentheses open parentheses square root of 29 close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of 1.015 end root end fraction end cell row blank left right double arrow cell theta equals cos to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 1.015 end root end fraction close parentheses equals 88 comma 2 degree end cell end table

Dengan demikian, besar sudut yang dibentuk oleh vektor u with rightwards arrow on top equals 5 i with hat on top plus j with hat on top minus 3 k with hat on top dan vektor v with rightwards arrow on top equals 2 i with hat on top plus 3 j with hat on top plus 4 k with hat on top adalah 88 comma 2 degree.

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