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Diketahui vektor , dan, ,maka besar sudut antara vektor , dan vektor , adalah ....

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PERTANYAAN :

Diketahui vektor begin mathsize 14px style straight a with rightwards harpoon with barb upwards on top equals straight i with rightwards harpoon with barb upwards on top plus 3 straight j with rightwards harpoon with barb upwards on top plus 2 straight k with rightwards harpoon with barb upwards on top end style dan begin mathsize 14px style straight b with rightwards harpoon with barb upwards on top equals 3 straight i with rightwards harpoon with barb upwards on top plus 2 straight j with rightwards harpoon with barb upwards on top minus straight k with rightwards harpoon with barb upwards on top end style, maka besar sudut antara vektor begin mathsize 14px style straight a with rightwards harpoon with barb upwards on top end style dan vektor begin mathsize 14px style straight b with rightwards harpoon with barb upwards on top end style adalah ….

Jawaban

 besar sudut antara vektor begin mathsize 14px style straight a with rightwards harpoon with barb upwards on top end style dan vektor begin mathsize 14px style straight b with rightwards harpoon with barb upwards on top end style adalah begin mathsize 14px style 60 degree end style.


Jika begin mathsize 14px style straight alpha end style adalah sudut antara vektor vektor begin mathsize 14px style a with rightwards harpoon with barb upwards on top end style dan begin mathsize 14px style b with rightwards harpoon with barb upwards on top end style, maka nilai begin mathsize 14px style straight alpha end style dapat ditentukan dengan: 

begin mathsize 14px style cos space straight alpha equals fraction numerator a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top over denominator open vertical bar a with rightwards harpoon with barb upwards on top close vertical bar open vertical bar b with rightwards harpoon with barb upwards on top close vertical bar end fraction end style

Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space straight alpha end cell equals cell fraction numerator open parentheses table row 1 row 3 row 2 end table close parentheses times open parentheses table row 3 row 2 row cell negative 1 end cell end table close parentheses over denominator square root of 1 squared plus 3 squared plus 2 squared end root square root of 3 squared plus 2 squared plus open parentheses negative 1 close parentheses squared end root end fraction end cell row cell cos space straight alpha end cell equals cell fraction numerator 1 times 3 plus 3 times 2 plus 2 times open parentheses negative 1 close parentheses over denominator square root of 1 plus 9 plus 4 end root square root of 9 plus 4 plus 1 end root end fraction end cell row cell cos space straight alpha end cell equals cell fraction numerator 3 plus 6 minus 2 over denominator square root of 14 square root of 14 end fraction end cell row cell cos space straight alpha end cell equals cell 7 over 14 end cell row cell cos space straight alpha end cell equals cell 1 half end cell row straight alpha equals cell arc space cos open parentheses 1 half close parentheses end cell row straight alpha equals cell 60 degree end cell end table  

Jadi, besar sudut antara vektor begin mathsize 14px style straight a with rightwards harpoon with barb upwards on top end style dan vektor begin mathsize 14px style straight b with rightwards harpoon with barb upwards on top end style adalah begin mathsize 14px style 60 degree end style.


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