Diketahui koordinat, ,, , , . Jika, merupakan vektor posisi dari, tentukan :, e. proyeksi skalar vektor, pada arah vektor, .

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PERTANYAAN :

Diketahui koordinat straight A left parenthesis negative 1 comma space 2 comma space 3 right parenthesisstraight B left parenthesis 4 comma space minus 5 comma space 0 right parenthesis, straight C left parenthesis negative 2 comma space minus 1 comma space 3 right parenthesis. Jika straight a with rightwards arrow on top comma straight b with rightwards arrow on top comma space dan space straight c with rightwards arrow on top spacemerupakan vektor posisi dari straight A comma straight B comma space dan space straight C comma tentukan : 

e. proyeksi skalar vektor open parentheses straight a with rightwards arrow on top minus straight b with rightwards arrow on top close parentheses pada arah vektor open parentheses straight b with rightwards arrow on top plus straight c with rightwards arrow on top close parentheses.

Table of Contents

Jawaban

jawaban yang benar adalah fraction numerator negative 43 over denominator 7 end fraction.


Pembahasan

Jawaban yang benar dari pertanyaan tersebut adalah fraction numerator negative 43 over denominator 7 end fraction.

Misalkan open parentheses straight a with rightwards arrow on top minus straight b with rightwards arrow on top close parentheses space equals straight p with rightwards arrow on top dan open parentheses straight b with rightwards arrow on top plus straight c with rightwards arrow on top close parentheses space equals straight q with rightwards arrow on top, maka diperoleh : 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell straight p with rightwards arrow on top end cell equals cell open parentheses straight a with rightwards arrow on top minus straight b with rightwards arrow on top close parentheses space end cell row blank equals cell open parentheses table row cell negative 1 end cell row 2 row 3 end table close parentheses minus open parentheses table row 4 row cell negative 5 end cell row 0 end table close parentheses end cell row blank equals cell open parentheses table row cell table row cell negative 1 minus 4 end cell row cell 2 minus left parenthesis negative 5 right parenthesis end cell end table end cell row cell 3 minus 0 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 5 end cell row cell 2 plus 5 end cell row 3 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 5 end cell row 7 row 3 end table close parentheses end cell row blank blank blank end table
  • table attributes columnalign right center left columnspacing 0px end attributes row cell straight q with rightwards arrow on top end cell equals cell open parentheses straight b with rightwards arrow on top plus straight c with rightwards arrow on top close parentheses space end cell row blank equals cell open parentheses table row 4 row cell negative 5 end cell row 0 end table close parentheses plus open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 3 end table close parentheses end cell row blank equals cell open parentheses table row cell 4 minus 2 end cell row cell negative 5 minus 1 end cell row cell 0 plus 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 2 row cell negative 6 end cell row 3 end table close parentheses end cell end table

Ingat proyeksi skalar straight a with rightwards arrow on top pada straight b with rightwards arrow on top dirumuskan : 

open vertical bar straight c with rightwards arrow on top close vertical bar equals fraction numerator straight a with rightwards arrow on top times straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction

sehingga proyeksi skalar vektor open parentheses straight a with rightwards arrow on top minus straight b with rightwards arrow on top close parentheses pada arah vektor open parentheses straight b with rightwards arrow on top plus straight c with rightwards arrow on top close parentheses yaitu : 

open vertical bar straight c with rightwards arrow on top close vertical bar equals fraction numerator straight p with rightwards arrow on top times straight q with rightwards arrow on top over denominator open vertical bar straight q with rightwards arrow on top close vertical bar end fraction

perhatikan beberapa sifat berikut : 

  • perkalian titik pada vektor jika straight a with rightwards arrow on top equals straight x subscript 1 straight i with rightwards arrow on top plus straight y subscript 1 straight j with rightwards harpoon with barb upwards on top plus straight z subscript 1 straight k with rightwards arrow on top dan straight b with rightwards arrow on top equals straight x subscript 2 straight i with rightwards arrow on top plus straight y subscript 2 straight j with rightwards harpoon with barb upwards on top plus straight z subscript 2 straight k with rightwards arrow on top maka straight a with rightwards arrow on top times straight b with rightwards arrow on top equals straight x subscript 1 straight x subscript 2 plus straight y subscript 1 straight y subscript 2 plus straight z subscript 1 straight z subscript 2 
  • open vertical bar straight a with rightwards arrow on top close vertical bar equals square root of straight x subscript 1 squared plus straight y subscript 1 squared plus straight z subscript 1 squared end root

sehingga diperoleh : 

table attributes columnalign right center left columnspacing 0px end attributes row cell straight p with rightwards arrow on top times straight q with rightwards arrow on top end cell equals cell open parentheses negative 5 close parentheses open parentheses 2 close parentheses plus open parentheses 7 close parentheses open parentheses negative 6 close parentheses plus open parentheses 3 close parentheses open parentheses 3 close parentheses end cell row blank equals cell negative 10 minus 42 plus 9 end cell row blank equals cell negative 43 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar q with rightwards arrow on top close vertical bar end cell equals cell square root of 2 squared plus open parentheses negative 6 close parentheses squared plus 3 squared end root end cell row blank equals cell square root of 4 plus 36 plus 9 end root end cell row blank equals cell square root of 49 end cell row blank equals 7 end table

substitusikan nilai yang diketahui pada rumus sehingga menjadi seperti berikut : 

open vertical bar straight c with rightwards arrow on top close vertical bar equals fraction numerator straight p with rightwards arrow on top times straight q with rightwards arrow on top over denominator open vertical bar straight q with rightwards arrow on top close vertical bar end fraction open vertical bar straight c with rightwards arrow on top close vertical bar equals fraction numerator negative 43 over denominator 7 end fraction

Dengan demikian, jawaban yang benar adalah fraction numerator negative 43 over denominator 7 end fraction.


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