Jarak titik ( sama dengan 10. Dengan demikian nilai

Pertanyaan

Jarak titik ( $begin mathsize 14px style P open parentheses 6 comma space minus 3 comma space 5 close parentheses space dan space Q open parentheses negative 2 comma space 3 comma space c close parentheses end style$ sama dengan 10. Dengan demikian nilai $size 14px c size 14px equals size 14px. size 14px. size 14px. size 14px.$

Pembahasan Video:

Pembahasan Soal:

Dengan menggunakan rumus panjang vektor 3 dimensi, maka:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row jarak equals cell square root of open parentheses negative 2 minus 6 close parentheses squared plus open parentheses 3 minus left parenthesis negative 3 right parenthesis close parentheses squared plus open parentheses c minus 5 close parentheses squared end root end cell row 10 equals cell square root of open parentheses negative 8 close parentheses squared plus 6 squared plus open parentheses c minus 5 close parentheses squared end root end cell row 100 equals cell 100 plus open parentheses c minus 5 close parentheses squared end cell row cell open parentheses c minus 5 close parentheses squared end cell equals 0 row cell c minus 5 end cell equals 0 row c equals 5 end table end style$

Jadi, nilai dari c adalah 5.

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Dijawab oleh:

D. Enty

Terakhir diupdate 12 Juli 2021

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Tentukan modulus dari vektor-vektor berikut. b. dengan titik dan titik B

Pembahasan Soal:

Vektor $begin mathsize 14px style stack A B with rightwards arrow on top end style$ dapat ditentukan sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack A B with rightwards arrow on top end cell equals cell B minus A end cell row blank equals cell open parentheses table row 2 row 1 row cell negative 4 end cell end table close parentheses minus open parentheses table row cell negative 2 end cell row 3 row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row 4 row cell negative 2 end cell row cell negative 3 end cell end table close parentheses end cell end table end style$

Modulus vektor tersebut dapat ditentukan sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack A B with rightwards arrow on top close vertical bar end cell equals cell square root of x squared plus y squared plus z squared end root end cell row blank equals cell square root of 4 squared plus open parentheses negative 2 close parentheses squared plus open parentheses negative 3 close parentheses squared end root end cell row blank equals cell square root of 16 plus 4 plus 9 end root end cell row blank equals cell square root of 29 end cell end table end style$

Dengan demikian, modulus vektor $begin mathsize 14px style stack A B with rightwards arrow on top end style$ adalah $begin mathsize 14px style square root of 29 end style$

Roboguru

Tentukan modulus dari vektor-vektor berikut. b. dengan titik dan titik .

Pembahasan Soal:

Modulus vektor adalah besar atau nilai vektor. Jika diketahui titik $begin mathsize 14px style straight A open parentheses x subscript 1 comma space y subscript 1 comma space z subscript 1 close parentheses end style$ dan titik $begin mathsize 14px style straight B open parentheses x subscript 2 comma space y subscript 2 comma space z subscript 2 close parentheses end style$ , maka besar nilai vektor $begin mathsize 14px style AB with rightwards arrow on top end style$ adalah

$begin mathsize 14px style AB with rightwards arrow on top equals square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared plus open parentheses z subscript 2 minus z subscript 1 close parentheses squared end root end style$ .

Akan ditentukan modulus dari $begin mathsize 14px style AB with rightwards arrow on top end style$ dengan titik $begin mathsize 14px style straight A open parentheses negative 2 comma space 3 comma space minus 1 close parentheses end style$ dan titik $begin mathsize 14px style straight B open parentheses 2 comma space 1 comma space minus 4 close parentheses end style$ :

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell AB with rightwards arrow on top end cell equals cell square root of open parentheses 2 minus open parentheses negative 2 close parentheses close parentheses squared plus open parentheses 1 minus 3 close parentheses squared plus open parentheses negative 4 minus open parentheses negative 1 close parentheses close parentheses squared end root end cell row blank equals cell square root of 4 squared plus open parentheses negative 2 close parentheses squared plus open parentheses negative 3 close parentheses squared end root end cell row blank equals cell square root of 16 plus 4 plus 9 end root end cell row cell AB with rightwards arrow on top end cell equals cell square root of 29 end cell end table end style$

Jadi, diperoleh modulus dari vektor tersebut adalah $begin mathsize 14px style square root of 29 end style$ .

Roboguru

Tentukan modulus dari vektor-vektor berikut. b. dengan titik dan titik .

Pembahasan Soal:

$begin mathsize 14px style stack A B with rightwards arrow on top equals b with rightwards arrow on top minus a with rightwards arrow on top end style$

$begin mathsize 14px style equals open parentheses table row 2 row 1 row cell negative 4 end cell end table close parentheses minus open parentheses table row cell negative 2 end cell row 3 row cell negative 1 end cell end table close parentheses equals open parentheses table row 4 row cell negative 2 end cell row cell negative 3 end cell end table close parentheses end style$

Modulus artinya panjang vektor.

$begin mathsize 14px style open vertical bar stack A B with rightwards arrow on top close vertical bar equals square root of 4 squared plus open parentheses negative 2 close parentheses squared plus open parentheses negative 3 close parentheses squared space end root equals square root of 29 end style$

Jadi modulus vektor AB adalah 29.

Roboguru

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

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

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

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.

Roboguru

Diketahui vektor . Tentukan: a.

Pembahasan Soal:

$begin mathsize 14px style a with rightwards arrow on top plus b with rightwards arrow on top end style$ didapatkan:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top plus b with rightwards arrow on top end cell equals cell open parentheses table row 0 row cell negative 1 end cell row 3 end table close parentheses plus open parentheses table row cell negative 2 end cell row 0 row 1 end table close parentheses end cell row cell a with rightwards arrow on top plus b with rightwards arrow on top end cell equals cell open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 4 end table close parentheses end cell end table end style$

$begin mathsize 14px style open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar end style$ didapatkan:

$begin mathsize 14px style open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar equals square root of open parentheses negative 2 close parentheses squared plus open parentheses negative 1 close parentheses squared plus 4 squared end root open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar equals square root of 4 plus 1 plus 16 end root open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar equals square root of 21 end style$

jadi, hasil dari $open vertical bar a with rightwards arrow on top plus b with rightwards arrow on top close vertical bar space adalah space square root of 21$

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