Pertanyaan

Diketahui a = 2 i + 2 j − k dan b = 6 i − 3 j + 2 k . Tentukan panjang proyeksi dan vektor proyeksi a terhadap vektor b.

Diketahui $a = 2 i + 2 j - k$ dan $b = 6 i - 3 j + 2 k$ . Tentukan panjang proyeksi dan vektor proyeksi a terhadap vektor b.

H. Firmansyah

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

panjang proyeksi dan vektor preyeksi aterhadap bberturut-turut adalah .

panjang proyeksi dan vektor preyeksi a terhadap b berturut-turut adalah 4 over 7 space dan space 24 over 49 straight i minus 12 over 49 straight j plus 8 over 49 straight k

4 over 7 space dan space 24 over 49 straight i minus 12 over 49 straight j plus 8 over 49 straight k

Pembahasan

Ingat bahwa, rumus panjang proyeksi dan vektor preyeksi u terhadap v berturut-turut adalah maka, panjang proyeksi a terhadap vektor b adalah dan proyeksi vektor a terhadap vektor b adalah Oleh karena itu, panjang proyeksi dan vektor preyeksi aterhadap bberturut-turut adalah .

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 a terhadap vektor b adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses over denominator square root of 6 squared plus left parenthesis negative 3 right parenthesis squared plus 2 to the power of 2 end exponent end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 12 minus 6 minus 2 over denominator square root of 36 plus 9 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 4 over denominator square root of 49 end fraction close vertical bar end cell row blank equals cell open vertical bar 4 over 7 close vertical bar end cell row blank equals cell 4 over 7 end cell end table$

dan proyeksi vektor a terhadap vektor b adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell x with rightwards arrow on top end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar squared end fraction times b with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 2 row 2 row cell negative 1 end cell end table close parentheses times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses over denominator open square brackets square root of 6 squared plus left parenthesis negative 3 right parenthesis squared plus 2 squared end root close square brackets squared end fraction times open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell 4 over 49 open parentheses table row 6 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell open parentheses table row cell 24 over 49 end cell row cell negative 12 over 49 end cell row cell 8 over 49 end cell end table close parentheses end cell end table$

Oleh karena itu, panjang proyeksi dan vektor preyeksi a terhadap b berturut-turut adalah 4 over 7 space dan space 24 over 49 straight i minus 12 over 49 straight j plus 8 over 49 straight k .