Pertanyaan

Diketahui P ( 2 , 4 , 3 ) dan Q ( 1 , − 5 , 2 ) . O adalah titik pangkal. Tentukan panjang proyeksi dan vektor proyeksi qke p.

Diketahui $P (2, 4, 3)$ dan $Q (1, - 5, 2)$ . O adalah titik pangkal. Tentukan panjang proyeksi dan vektor proyeksi q ke p.

F. Kurnia

Master Teacher

Mahasiswa/Alumni Universitas Jember

Jawaban terverifikasi

Jawaban

panjang proyeksi dan vektor preyeksi qterhadap pberturut-turut adalah .

panjang proyeksi dan vektor preyeksi q terhadap p berturut-turut adalah 12 over 29 square root of 29 space dan space minus 14 over 29 straight i minus 48 over 29 straight j minus 36 over 29 straight k

12 over 29 square root of 29 space dan space minus 14 over 29 straight i minus 48 over 29 straight j minus 36 over 29 straight k

Pembahasan

Diketahui dan , maka Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah maka, panjang proyeksi qterhadap vektor padalah dan proyeksi vektor qterhadap vektor padalah Oleh karena itu, panjang proyeksi dan vektor preyeksi qterhadap pberturut-turut adalah .

Diketahui P left parenthesis 2 comma space 4 comma space 3 right parenthesis dan Q left parenthesis 1 comma space minus 5 comma space 2 right parenthesis , maka

straight p equals 2 straight i plus 4 straight j plus 3 straight k space dan space straight q equals straight i minus 5 straight j plus 2 straight k

Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah

$open vertical bar x with rightwards arrow on top close vertical bar equals open vertical bar fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close vertical bar space dan space x with rightwards arrow on top equals fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction times v with rightwards arrow on top$

maka, panjang proyeksi q terhadap vektor p adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar p with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 2 squared plus 4 squared plus 3 squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 2 minus 20 plus 6 over denominator square root of 4 plus 16 plus 9 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 12 over denominator square root of 29 end fraction close vertical bar end cell row blank equals cell 12 over 29 square root of 29 end cell end table$

dan proyeksi vektor q terhadap vektor p adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar p with rightwards arrow on top close vertical bar squared end fraction times p with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 2 squared plus 4 squared plus 3 squared end root squared end fraction times open parentheses table row 2 row 4 row 3 end table close parentheses end cell row blank equals cell fraction numerator 2 minus 20 plus 6 over denominator 29 end fraction times open parentheses table row 2 row 4 row 3 end table close parentheses end cell row blank equals cell fraction numerator negative 12 over denominator 29 end fraction times open parentheses table row 2 row 4 row 3 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 14 over 29 end cell row cell negative 48 over 29 end cell row cell negative 36 over 29 end cell end table close parentheses end cell end table$

Oleh karena itu, panjang proyeksi dan vektor preyeksi q terhadap p berturut-turut adalah 12 over 29 square root of 29 space dan space minus 14 over 29 straight i minus 48 over 29 straight j minus 36 over 29 straight k .