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Diketahui vektor dan . Nilai sinus sudut antara ve...

Diketahui vektor begin mathsize 14px style a space left parenthesis 2 comma negative 3 comma 1 right parenthesis end style dan begin mathsize 14px style b space left parenthesis 1 comma negative 2 comma 3 right parenthesis end style. Nilai sinus sudut antara vektor begin mathsize 14px style a space dan space b end style adalah ....

Jawaban:

Nilai sinus antara dua vektor diperoleh dengan memanfaatkan rumus dot product begin mathsize 14px style a times b equals open vertical bar a close vertical bar open vertical bar b close vertical bar cos space theta end style, dicari niliai cosinusnya terlebih dahulu.

Diketahui:

begin mathsize 14px style a equals left parenthesis 2 comma negative 3 comma 1 right parenthesis b equals left parenthesis 1 comma negative 2 comma 3 right parenthesis end style
Misal, begin mathsize 14px style theta equals end stylesudut antara kedua vektor

Ditanya: begin mathsize 14px style sin space theta end style?

Jawab:

Akan dicari nilai begin mathsize 14px style cos space theta end style terlebih dahulu.
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space theta end cell equals cell fraction numerator a times b over denominator open vertical bar a close vertical bar open vertical bar b close vertical bar end fraction end cell row blank equals cell fraction numerator left parenthesis 2 times 1 right parenthesis plus left parenthesis left parenthesis negative 3 right parenthesis times left parenthesis negative 2 right parenthesis right parenthesis plus left parenthesis 1 times 3 right parenthesis over denominator square root of 2 squared plus left parenthesis negative 3 right parenthesis squared plus 1 to the power of 2 end exponent end root times square root of 1 squared plus left parenthesis negative 2 right parenthesis squared plus 3 to the power of 2 end exponent end root end fraction end cell row blank equals cell fraction numerator 2 plus 6 plus 3 over denominator square root of 4 plus 9 plus 1 end root times square root of 1 plus 4 plus 9 end root end fraction end cell row blank equals cell fraction numerator 11 over denominator square root of 14 times square root of 14 end fraction end cell row blank equals cell 11 over 14 end cell end table end style
 

Didapat nilai begin mathsize 14px style cos space theta equals 11 over 14 end style.

Memanfaatkan perbandingan sisi-sisi segitiga begin mathsize 14px style cos space theta equals x over r end style dan begin mathsize 14px style sin space theta equals y over r end style akan dicari nilai sinusnya.

Nilai begin mathsize 14px style x equals 11 space dan space r equals 14 end style
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell square root of r squared minus x squared end root end cell row blank equals cell square root of 14 squared minus 11 squared end root end cell row blank equals cell square root of 196 minus 121 end root end cell row blank equals cell square root of 75 end cell row blank equals cell 5 square root of 3 end cell end table end style
 

Nilai sinus sudut antara kedua vektor:
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space theta end cell equals cell y over r end cell row blank equals cell fraction numerator 5 square root of 3 over denominator 14 end fraction end cell row blank equals cell 5 over 14 square root of 3 end cell end table end style
 

Dengan demikian, Nilai sinus sudut antara vektor begin mathsize 14px style a space dan space b end style adalah begin mathsize 14px style 5 over 14 square root of 3 end style.

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