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Diketahui vector u ⇀ = 2 i ⇀ + p j ⇀ ​ + 2 k ⇀ dan v ⇀ = 3 i ⇀ − j ⇀ ​ + k ⇀ .Jika proyeksi scalar orthogonal vector v ⇀ pada arah u ⇀ sama dengan setengah panjang vector , maka nilai adalah ....

Diketahui vector  dan . Jika proyeksi scalar orthogonal vector  pada arah  sama dengan setengah panjang vector u with rightwards harpoon with barb upwards on top, maka nilai p adalah ....

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F. Nur

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Mahasiswa/Alumni Universitas Muhammadiyah Malang

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nilai atau .

nilai p equals negative 4 atau p equals 2

Pembahasan

Diketahui: Sehingga, panjang proyeksi orthogonal pada arah : Jadi, nilai atau .

Diketahui:

  • u with rightwards harpoon with barb upwards on top equals 2 i with rightwards harpoon with barb upwards on top plus p j with rightwards harpoon with barb upwards on top plus 2 k with rightwards harpoon with barb upwards on top 
  • v with rightwards harpoon with barb upwards on top equals 3 i with rightwards harpoon with barb upwards on top minus j with rightwards harpoon with barb upwards on top plus space k with rightwards harpoon with barb upwards on top 

Sehingga, panjang proyeksi orthogonal v with rightwards harpoon with barb upwards on top pada arah u with rightwards harpoon with barb upwards on top:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator v with rightwards harpoon with barb upwards on top times u with rightwards harpoon with barb upwards on top over denominator open vertical bar u with rightwards harpoon with barb upwards on top close vertical bar end fraction end cell equals cell 1 half open vertical bar u with rightwards arrow on top close vertical bar end cell row cell fraction numerator 3 times 2 plus open parentheses negative 1 close parentheses times p plus 1 times 2 over denominator open vertical bar u with rightwards arrow on top close vertical bar end fraction end cell equals cell 1 half open vertical bar u with rightwards arrow on top close vertical bar end cell row cell fraction numerator 6 minus p plus 2 over denominator open vertical bar u with rightwards arrow on top close vertical bar end fraction end cell equals cell 1 half open vertical bar u with rightwards arrow on top close vertical bar end cell row cell 8 minus p end cell equals cell 1 half open vertical bar u with rightwards arrow on top close vertical bar squared end cell row cell 8 minus p end cell equals cell 1 half open parentheses 2 squared plus p squared plus 2 squared close parentheses end cell row cell 8 minus p end cell equals cell 1 half open parentheses 4 plus p squared plus 4 close parentheses end cell row cell 2 open parentheses 8 minus p close parentheses end cell equals cell 8 plus p squared end cell row cell 16 minus 2 p end cell equals cell 8 plus p squared end cell row cell negative p squared minus 2 p plus 8 end cell equals 0 row cell p squared plus 2 p minus 8 end cell equals 0 row cell open parentheses p plus 4 close parentheses open parentheses p minus 2 close parentheses end cell equals 0 row blank blank cell p equals negative 4 space atau space p equals 2 end cell end table 

  Jadi, nilai p equals negative 4 atau p equals 2

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