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Diketahui vektor  dan vektor . Proyeksi vektor  pa...

Diketahui vektor a equals 2 i plus j plus 3 k dan vektor b equals i plus 3 j minus p k. Proyeksi vektor a pada vektor b adalah 1 half i plus 3 over 2 j minus 1 half p k, tentukan nilai p.space

Jawaban:

Perhatikan gambar berikut.



 

Proyeksi vektor ortogonal a with rightwards arrow on top pada b with rightwards arrow on top dapat dirumuskan:

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

dengan:

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell proyeksi space vektor space ortogonal space a with rightwards arrow on top space pada space b with rightwards arrow on top end cell row cell open vertical bar b with rightwards arrow on top close vertical bar end cell equals cell panjang space vektor space b with rightwards arrow on top end cell end table  

Panjang vektor b with rightwards arrow on top dapat dirumuskan:

open vertical bar b with rightwards arrow on top close vertical bar equals square root of b subscript 1 squared plus b subscript 2 squared end root 

dengan:

table attributes columnalign right center left columnspacing 0px end attributes row cell b subscript 1 end cell equals cell elemen space vektor space b with rightwards arrow on top space pada space sumbu space x end cell row cell b subscript 2 end cell equals cell elemen space vektor space stack b space with rightwards arrow on top pada space sumbu space y end cell end table 

Diketahui: 

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell 2 i with rightwards arrow on top plus j with rightwards arrow on top plus 3 k with rightwards arrow on top end cell row cell b with rightwards arrow on top end cell equals cell i with rightwards arrow on top plus 3 j with rightwards arrow on top minus p k with rightwards arrow on top end cell row cell c with rightwards arrow on top end cell equals cell 1 half i with rightwards arrow on top plus 3 over 2 j with rightwards arrow on top minus 1 half p k with rightwards arrow on top end cell end table  

Gunakan rumus proyeksi vektor ortogonal agar diperoleh kesamaan matriks.

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar squared end fraction times b with rightwards arrow on top end cell row cell open parentheses table row cell 1 half end cell row cell 3 over 2 end cell row cell negative 1 half p end cell end table close parentheses end cell equals cell fraction numerator open parentheses table row 2 row 1 row 3 end table close parentheses times open parentheses table row 1 row 3 row cell negative p end cell end table close parentheses over denominator open parentheses square root of 1 squared plus 3 squared plus open parentheses negative p close parentheses squared end root close parentheses squared end fraction times open parentheses table row 1 row 3 row cell negative p end cell end table close parentheses end cell row cell open parentheses table row cell 1 half end cell row cell 3 over 2 end cell row cell negative 1 half p end cell end table close parentheses end cell equals cell fraction numerator 5 minus 3 p over denominator 10 plus p squared end fraction times open parentheses table row 1 row 3 row cell negative p end cell end table close parentheses end cell row cell open parentheses table row cell 1 half end cell row cell 3 over 2 end cell row cell negative 1 half p end cell end table close parentheses end cell equals cell open parentheses table row cell fraction numerator 5 minus 3 p over denominator 10 plus p squared end fraction end cell row cell 3 times fraction numerator 5 minus 3 p over denominator 10 plus p squared end fraction end cell row cell negative p times fraction numerator 5 minus 3 p over denominator 10 plus p squared end fraction end cell end table close parentheses end cell end table 

Diperoleh 3 persamaan. Maka gunakan persamaan pertama untuk menentukan nilai p.

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 half end cell equals cell fraction numerator 5 minus 3 p over denominator 10 plus p squared end fraction end cell row cell 10 plus p squared end cell equals cell 2 open parentheses 5 minus 3 p close parentheses end cell row cell 10 plus p squared end cell equals cell 10 minus 6 p end cell row cell p squared plus 6 p end cell equals 0 row cell p open parentheses p plus 6 close parentheses end cell equals 0 row p equals cell 0 logical or p equals negative 6 end cell end table 

Jadi, p equals 0 atau p equals negative 6.

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