Diketahui vektor dan dan panjang proyeksi vektor pada vektor adalah . Sudut antara dan adalah , maka

Pertanyaan

Diketahui vektor $stack text a end text with rightwards arrow on top equals open parentheses table row x row 1 row 2 end table close parentheses$ dan $stack text b end text with rightwards arrow on top equals open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses$ dan panjang proyeksi vektor $stack text a end text with rightwards arrow on top$ pada vektor $stack text b end text with rightwards arrow on top$ adalah $1 over 7 square root of 14$ . Sudut antara $stack text a end text with rightwards arrow on top$ dan $stack text b end text with rightwards arrow on top$ adalah $alpha$ , maka $cos space alpha equals space...$

$1 over 7 square root of 7$
$1 over 14 square root of 14$
$1 over 21 square root of 21$
$1 over 27 square root of 27$
$1 over 28 square root of 28$

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

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

Jika vektor $a with rightwards arrow on top$ pada vektor $b with rightwards arrow on top$ membentuk sudut $alpha$ , maka berlaku:

$a with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha$

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

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

Berdasarkan konsep di atas, nilai $x$ dapat ditentukan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar c with rightwards arrow on top close vertical bar end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator open parentheses table row x row 1 row 2 end table close parentheses open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses over denominator square root of 2 squared plus 3 squared plus open parentheses negative 1 close parentheses squared end root end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator 2 x plus 1 times 3 plus 2 open parentheses negative 1 close parentheses over denominator square root of 4 plus 9 plus 1 end root end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator 2 x plus 1 over denominator square root of 14 end fraction end cell row cell 1 over 7 square root of 14 times square root of 14 end cell equals cell 2 x plus 1 end cell row cell 14 over 7 end cell equals cell 2 x plus 1 end cell row 2 equals cell 2 x plus 1 end cell row cell 2 x end cell equals 1 row x equals cell 1 half end cell end table$

Diperoleh $x equals 1 half$ sehingga $a with rightwards arrow on top equals open parentheses table row cell 1 half end cell row 1 row 2 end table close parentheses$

Nilai $cos space alpha$ adalah sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator open vertical bar a with rightwards arrow on top close vertical bar end fraction times fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of open parentheses begin display style 1 half end style close parentheses squared plus 1 squared plus 2 squared end root end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 1 over denominator square root of begin display style 21 over 4 end style end root end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 1 over denominator begin display style 1 half end style square root of 21 end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 2 over denominator square root of 21 end fraction times 1 over 7 square root of 14 end cell row blank equals cell 2 over 21 square root of 6 end cell end table$

Oleh karena itu, tidak terdapat pilihan jawaban yang tepat.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Eka

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 11 Juli 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Jika dan maka proyeksi skalar orthogonal vektor pada = ...

Pembahasan Soal:

Ingat kembali proyeksi skalar orthogonal berikut.

Jika vektor $top enclose a$ diproyeksikan secara orthogonal pada $top enclose b$ , maka proyeksi skalar orthogonalnya adalah

$open vertical bar top enclose c close vertical bar equals open vertical bar fraction numerator top enclose a bullet top enclose b over denominator open vertical bar top enclose b close vertical bar end fraction close vertical bar$

dengan

$top enclose a bullet top enclose b equals a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3 space dan space open vertical bar top enclose b close vertical bar equals square root of b subscript 1 squared plus b subscript 2 squared plus b subscript 3 squared end root$

Dari rumus di atas, diperoleh perhitungan sebagai berikut.

Menentukan $top enclose a bullet top enclose b$

$table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose a bullet top enclose b end cell equals cell 3 left parenthesis 6 right parenthesis plus 1 left parenthesis 2 right parenthesis plus left parenthesis negative 2 right parenthesis left parenthesis 3 right parenthesis end cell row blank equals cell 18 plus 2 minus 6 end cell row blank equals 14 end table$

Menentukan $open vertical bar top enclose b close vertical bar$

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar top enclose b close vertical bar end cell equals cell square root of 6 squared plus 2 squared plus 3 squared end root end cell row blank equals cell square root of 36 plus 4 plus 9 end root end cell row blank equals cell square root of 49 end cell row blank equals 7 end table$

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$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar top enclose c close vertical bar end cell equals cell open vertical bar fraction numerator top enclose a bullet top enclose b over denominator open vertical bar top enclose b close vertical bar end fraction close vertical bar end cell row blank equals cell vertical line 14 over 7 vertical line end cell row blank equals cell vertical line 2 vertical line end cell row blank equals 2 end table$

Jadi, proyeksi skalar orthogonal vektor $top enclose a$ pada $top enclose b$ adalah 2.

Oleh karena itu, jawaban yang benar adalah B.

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Diketahui titik-titik , dan . Jika titik merupakan proyeksi titik pada garis maka panjang sama dengan ...

Pembahasan Soal:

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

Panjang proyeksi vektor $stack text a end text with rightwards arrow on top$ pada vektor $stack text b end text with rightwards arrow on top$ dapat ditentukan menggunakan rumus berikut.

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

Komponen vektor $stack A B with rightwards arrow on top$ dan $stack A C with rightwards arrow on top$ pada soal di atas dapat ditentukan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell stack A B with rightwards arrow on top end cell equals cell text B-A end text end cell row blank equals cell open parentheses table row 10 row 1 row 3 end table close parentheses minus open parentheses table row 2 row 1 row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 8 row 0 row 6 end table close parentheses end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell stack A C with rightwards arrow on top end cell equals cell text C-A end text end cell row blank equals cell open parentheses table row 8 row 4 row cell negative 1 end cell end table close parentheses minus open parentheses table row 2 row 1 row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 6 row 3 row 2 end table close parentheses end cell end table$

Berdasarkan konsep di atas, panjang proyeksi vektor $stack A C with rightwards arrow on top$ pada vektor $stack A B with rightwards arrow on top$ adalah sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar text AD end text close vertical bar end cell equals cell open vertical bar fraction numerator stack A C with rightwards arrow on top times stack A B with rightwards arrow on top over denominator open vertical bar stack A B with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row 6 row 3 row 2 end table close parentheses open parentheses table row 8 row 0 row 6 end table close parentheses over denominator square root of 8 squared plus 0 squared plus 6 squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 6 times 8 plus 3 times 0 plus 2 times 6 over denominator square root of 100 end fraction close vertical bar end cell row blank equals cell open vertical bar 60 over 10 close vertical bar end cell row blank equals 6 end table$

Diperoleh panjang $text AD=6 end text$

Oleh karena itu, jawaban yang tepat adalah C.

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Diketahui vektor dengan panjang dan . Jika panjang proyeksi vektor pada vektor sama dengan , maka nilai

Pembahasan Soal:

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

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

Berdasarkan konsep di atas, dapat ditentukan persamaan-persamaan berikut.

Persamaan 1:

Persamaan 2:

$table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 3 squared plus x squared plus y squared end root end cell equals cell square root of 14 end cell row cell 9 plus x squared plus y squared end cell equals 14 row cell x squared plus y squared end cell equals 5 end table$

Substitusi persamaan 1 ke persamaan 2 diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus y squared end cell equals 5 row cell open parentheses negative 2 y plus 5 close parentheses squared plus y squared end cell equals 5 row cell 4 y squared minus 20 y plus 25 plus y squared end cell equals 5 row cell 5 y squared minus 20 y plus 20 end cell equals 0 row cell y squared minus 4 y plus 4 end cell equals 0 row cell open parentheses y minus 2 close parentheses squared end cell equals 0 row y equals 2 end table$

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

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$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar c with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator open parentheses table row cell negative 2 end cell row 5 row 4 end table close parentheses open parentheses table row 4 row 5 row 7 end table close parentheses over denominator square root of 4 squared plus 5 squared plus 7 squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 2 times 4 plus 5 times 5 plus 4 times 7 over denominator square root of 16 plus 25 plus 49 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 8 plus 25 plus 28 over denominator square root of 90 end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 45 over denominator 3 square root of 10 end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 15 over denominator square root of 10 end fraction close vertical bar end cell row blank equals cell 15 over 10 square root of 10 end cell row blank equals cell 3 over 2 square root of 10 end cell end table$

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

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Berdasarkan konsep di atas, nilai $x$ dapat ditentukan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar c with rightwards arrow on top close vertical bar end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row 2 equals cell fraction numerator open parentheses table row 2 row cell negative 2 end cell row x end table close parentheses open parentheses table row cell negative 3 end cell row cell negative 4 end cell row 12 end table close parentheses over denominator square root of open parentheses negative 3 close parentheses squared plus open parentheses negative 4 close parentheses squared plus 12 squared end root end fraction end cell row 2 equals cell fraction numerator 2 open parentheses negative 3 close parentheses plus open parentheses negative 2 close parentheses open parentheses negative 4 close parentheses plus 12 x over denominator square root of 9 plus 16 plus 144 end root end fraction end cell row 2 equals cell fraction numerator negative 6 plus 8 plus 12 x over denominator square root of 169 end fraction end cell row 2 equals cell fraction numerator 2 plus 12 x over denominator 13 end fraction end cell row 26 equals cell 2 plus 12 x end cell row cell 12 x end cell equals 24 row x equals 2 end table$

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