Pertanyaan

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

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

F. Kurnia

Master Teacher

Mahasiswa/Alumni Universitas Jember

Jawaban terverifikasi

Jawaban

panjang proyeksi dan vektor preyeksi pterhadap qberturut-turut adalah .

panjang proyeksi dan vektor preyeksi p terhadap q berturut-turut adalah 2 over 5 square root of 30 space dan space minus 2 over 5 straight i plus 2 straight j minus 4 over 5 straight k

2 over 5 square root of 30 space dan space minus 2 over 5 straight i plus 2 straight j minus 4 over 5 straight k

Pembahasan

Diketahui dan , maka Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah maka, panjang proyeksi pterhadap vektor qadalah dan proyeksi vektor pterhadap vektor qadalah Oleh karena itu, panjang proyeksi dan vektor preyeksi pterhadap qberturut-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 p terhadap vektor q adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar q with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 1 squared plus left parenthesis negative 5 right parenthesis squared plus 2 to the power of 2 end exponent end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 2 minus 20 plus 6 over denominator square root of 1 plus 25 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 12 over denominator square root of 30 end fraction close vertical bar end cell row blank equals cell 2 over 5 square root of 30 end cell end table$

dan proyeksi vektor p terhadap vektor q adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell fraction numerator p with rightwards arrow on top times q with rightwards arrow on top over denominator open vertical bar q with rightwards arrow on top close vertical bar squared end fraction times q with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 2 row 4 row 3 end table close parentheses times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses over denominator square root of 1 squared plus left parenthesis negative 5 right parenthesis squared plus 2 squared end root squared end fraction times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses end cell row blank equals cell fraction numerator 2 minus 20 plus 6 over denominator 30 end fraction times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses end cell row blank equals cell fraction numerator negative 12 over denominator 30 end fraction times open parentheses table row 1 row cell negative 5 end cell row 2 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 over 5 end cell row 2 row cell negative 4 over 5 end cell end table close parentheses end cell end table$

Oleh karena itu, panjang proyeksi dan vektor preyeksi p terhadap q berturut-turut adalah 2 over 5 square root of 30 space dan space minus 2 over 5 straight i plus 2 straight j minus 4 over 5 straight k .