Vektor yang merupakan proyeksi vektor pada vektor adalah ...

Pertanyaan

Vektor yang merupakan proyeksi vektor $stack text x end text with rightwards arrow on top equals 3 stack text i end text with rightwards arrow on top plus stack text j end text with rightwards arrow on top minus stack text k end text with rightwards arrow on top$ pada vektor $stack text y end text with rightwards arrow on top equals 2 stack text i end text with rightwards arrow on top plus 5 stack text j end text with rightwards arrow on top plus stack text k end text with rightwards arrow on top$ adalah ...

$stack text i end text with rightwards arrow on top plus 5 over 2 stack text j end text with rightwards arrow on top plus 1 half stack text k end text with rightwards arrow on top$
$2 over 3 stack text i end text with rightwards arrow on top plus 5 over 3 stack text j end text with rightwards arrow on top plus 1 third stack text k end text with rightwards arrow on top$
$1 half stack text i end text with rightwards arrow on top plus 5 over 4 stack text j end text with rightwards arrow on top plus 1 fourth stack text k end text with rightwards arrow on top$
$2 over 5 stack text i end text with rightwards arrow on top minus stack text j end text with rightwards arrow on top minus 1 fifth stack text k end text with rightwards arrow on top$
$1 third stack text i end text with rightwards arrow on top plus 5 over 6 stack text j end text with rightwards arrow on top plus 1 over 6 stack text k end text with rightwards arrow on top$

Pembahasan Soal:

Panjang vektor $a with rightwards arrow on top equals x i with rightwards arrow on top plus y j with rightwards arrow on top plus z k with rightwards arrow on top$ , yaitu $open vertical bar a with rightwards arrow on top close vertical bar equals square root of x squared plus y squared plus z squared end root$

Jika vektor $a with rightwards arrow on top equals a subscript 1 i with rightwards arrow on top plus a subscript 2 j with rightwards arrow on top plus a subscript 3 k with rightwards arrow on top$ dan $b with rightwards arrow on top equals b subscript 1 i with rightwards arrow on top plus b subscript 2 j with rightwards arrow on top plus b subscript 3 k with rightwards arrow on top$ , maka

$a with rightwards arrow on top times b with rightwards arrow on top equals a subscript 1 times b subscript 1 plus a subscript 2 times b subscript 2 plus a subscript 3 times b subscript 3$

Proyeksi vektor orthogonal vektor $a with rightwards arrow on top$ pada $b with rightwards arrow on top$ dapat ditentukan menggunakan rumus berikut.

$c with rightwards arrow on top equals fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar squared end fraction times b with rightwards arrow on top$

Berdasarkan konsep tersebut dapat ditentukan penyelesaian soal sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell fraction numerator x with rightwards arrow on top times y with rightwards arrow on top over denominator open vertical bar y with rightwards arrow on top close vertical bar squared end fraction times y with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 3 row 1 row cell negative 1 end cell end table close parentheses open parentheses table row 2 row 5 row 1 end table close parentheses over denominator open parentheses square root of 2 squared plus 5 squared plus 1 end root close parentheses squared end fraction times open parentheses table row 2 row 5 row 1 end table close parentheses end cell row blank equals cell fraction numerator 3 times 2 plus 1 times 5 plus open parentheses negative 1 close parentheses times 1 over denominator open parentheses square root of 30 close parentheses squared end fraction times open parentheses table row 2 row 5 row 1 end table close parentheses end cell row blank equals cell 10 over 30 times open parentheses table row 2 row 5 row 1 end table close parentheses end cell row blank equals cell 1 third times open parentheses table row 2 row 5 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 over 3 end cell row cell 5 over 3 end cell row cell 1 third end cell end table close parentheses end cell end table$

Vektor proyeksi vektor $x with rightwards arrow on top$ pada $y with rightwards arrow on top$ adalah $2 over 3 i with rightwards arrow on top plus 5 over 3 j with rightwards arrow on top plus 1 third k with rightwards arrow on top$

Oleh karena itu, jawaban yang tepat adalah B.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Eka

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 11 Juli 2021

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Pembahasan Soal:

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Roboguru