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Diketahui vektor  dan  Jika panjang proyeksi vektor  pada  adalah  nilai

Pertanyaan

Diketahui vektor a with rightwards harpoon with barb upwards on top equals space 3 i space – space 4 j space plus space p k dan straight b with rightwards harpoon with barb upwards on top equals space 2 straight i space plus 2 straight j space minus 3 straight k. Jika panjang proyeksi vektor a with rightwards harpoon with barb upwards on top pada b with rightwards harpoon with barb upwards on top adalah fraction numerator 4 over denominator square root of 17 end fraction comma nilai p equals.... 

Pembahasan Soal:

Panjang proyeksi vektor a with rightwards harpoon with barb upwards on top pada b with rightwards harpoon with barb upwards on top  kita misalkan adalah open vertical bar straight P close vertical bar  ;

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open vertical bar straight P close vertical bar end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator open vertical bar straight a times straight b close vertical bar over denominator open vertical bar straight b close vertical bar end fraction end cell end table 

Dengan :

open vertical bar straight P close vertical bar : Panjang proyeksi vektor a with rightwards harpoon with barb upwards on top pada b with rightwards harpoon with barb upwards on top

open vertical bar straight b with rightwards harpoon with barb upwards on top close vertical bar : Panjang b with rightwards harpoon with barb upwards on top

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open vertical bar straight a times straight b close vertical bar end cell end table : Nilai mutlak dari table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight b end table 

Maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight P close vertical bar end cell equals cell fraction numerator open vertical bar straight a times straight b close vertical bar over denominator open vertical bar straight b close vertical bar end fraction end cell row cell fraction numerator 4 over denominator square root of 17 end fraction end cell equals cell fraction numerator open vertical bar left parenthesis 3 right parenthesis left parenthesis 2 right parenthesis plus left parenthesis negative 4 right parenthesis left parenthesis 2 right parenthesis plus left parenthesis straight p right parenthesis left parenthesis negative 3 right parenthesis close vertical bar over denominator square root of 2 squared plus 2 squared plus left parenthesis negative 3 right parenthesis squared end root end fraction end cell row cell fraction numerator 4 over denominator square root of 17 end fraction end cell equals cell fraction numerator open vertical bar 6 minus 8 minus 3 straight p close vertical bar over denominator square root of 4 plus 4 plus 9 end root end fraction end cell row cell fraction numerator 4 over denominator square root of 17 end fraction end cell equals cell fraction numerator open vertical bar negative 2 minus 3 straight p close vertical bar over denominator square root of 17 end fraction end cell row 4 equals cell open vertical bar negative 2 minus 3 straight p close vertical bar end cell row cell untuk space open vertical bar negative 2 minus 3 straight p close vertical bar end cell equals cell negative 2 minus 3 straight p end cell row 4 equals cell negative 2 minus 3 straight p end cell row 6 equals cell negative 3 straight p end cell row straight p equals cell negative 2 end cell row cell untuk space open vertical bar negative 2 minus 3 straight p close vertical bar end cell equals cell negative left parenthesis negative 2 minus 3 straight p right parenthesis end cell row 4 equals cell negative left parenthesis negative 2 minus 3 straight p right parenthesis end cell row 4 equals cell 2 plus 3 straight p end cell row 2 equals cell 3 straight p end cell row straight p equals cell 2 over 3 end cell end table 

Jadiii, Nilai p yang memenuhi adalah p equals negative 2 atau p equals 2 over 3. 

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 04 Mei 2021

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