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Diketahui vektor

Pertanyaan

Diketahui vektor straight p with rightwards harpoon with barb upwards on top equals open parentheses table row 2 row 5 end table close parentheses space dan space straight q with rightwards harpoon with barb upwards on top equals open parentheses table row 2 row 4 end table close parentheses space Tentukan space nilai space dari space 3 open parentheses straight p with rightwards harpoon with barb upwards on top plus straight q with rightwards harpoon with barb upwards on top close parentheses equals...

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 open parentheses straight p with rightwards harpoon with barb upwards on top plus straight q with rightwards harpoon with barb upwards on top close parentheses end cell equals cell 3 open parentheses table row cell 2 plus 2 end cell row cell 5 plus 4 end cell end table close parentheses end cell row blank equals cell 3 open parentheses table row 4 row 9 end table close parentheses end cell row blank equals cell open parentheses table row 12 row 27 end table close parentheses end cell end table

Jadi hasil dari table attributes columnalign right center left columnspacing 0px end attributes row cell 3 open parentheses straight p with rightwards harpoon with barb upwards on top plus straight q with rightwards harpoon with barb upwards on top close parentheses end cell equals cell open parentheses table row 12 row 27 end table close parentheses end cell end table

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Mahasiswa/Alumni UIN Walisongo Semarang

Terakhir diupdate 06 Juni 2021

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Pertanyaan yang serupa

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.

0

Roboguru

Diketahui A(3, 0, 0), B(0, - 3, 0), dan C(0, 0, 4). Panjang vektor proyeksi  ke vektor  adalah...

Pembahasan Soal:

Jika titik Pbegin mathsize 12px style open parentheses a subscript 1 comma space a subscript 2 comma space a subscript 3 close parentheses end style dan Qbegin mathsize 12px style open parentheses b subscript 1 comma space b subscript 2 comma space b subscript 3 close parentheses end style, maka:

begin mathsize 12px style stack P Q with rightwards arrow on top equals Q minus P equals open parentheses table row cell b subscript 1 end cell row cell b subscript 2 end cell row cell b subscript 3 end cell end table close parentheses minus open parentheses table row cell a subscript 1 end cell row cell a subscript 2 end cell row cell a subscript 3 end cell end table close parentheses end style

Misal begin mathsize 12px style P with rightwards arrow on top end stylebegin mathsize 12px style open parentheses a subscript 1 comma space a subscript 2 comma space a subscript 3 close parentheses end style dan begin mathsize 12px style Q with rightwards arrow on top end style = begin mathsize 12px style open parentheses b subscript 1 comma space b subscript 2 comma space b subscript 3 close parentheses end style, maka:

begin mathsize 12px style P with rightwards arrow on top. Q with rightwards arrow on top equals open parentheses a subscript 1 b subscript 1 plus a subscript 2 b subscript 2 plus a subscript 3 b subscript 3 close parentheses  open vertical bar P with rightwards arrow on top close vertical bar equals square root of open parentheses a subscript 1 close parentheses squared plus open parentheses a subscript 2 close parentheses squared plus open parentheses a subscript 3 close parentheses squared end root end style

Misal begin mathsize 12px style Z with rightwards arrow on top end style adalah proyeksi skalar vektor begin mathsize 12px style P with rightwards arrow on top end style pada vektor begin mathsize 12px style Q with rightwards arrow on top end style, maka:

begin mathsize 12px style Z with rightwards arrow on top equals fraction numerator P with rightwards arrow on top. Q with rightwards arrow on top over denominator stack open vertical bar Q close vertical bar with rightwards arrow on top end fraction space space end style

 

Diketahui A(3, 0, 0), B(0, - 3, 0), dan C(0, 0, 4). Panjang vektor proyeksi begin mathsize 12px style stack A C with rightwards arrow on top end style ke vektor begin mathsize 12px style stack A B with rightwards arrow on top end style, yaitu:

begin mathsize 12px style stack A C with rightwards arrow on top equals C minus A equals open parentheses table row 0 row 0 row 4 end table close parentheses minus open parentheses table row 3 row 0 row 0 end table close parentheses equals open parentheses table row cell negative 3 end cell row 0 row 4 end table close parentheses  stack A B with rightwards arrow on top equals B minus A equals open parentheses table row 0 row cell negative 3 end cell row 0 end table close parentheses minus open parentheses table row 3 row 0 row 0 end table close parentheses equals open parentheses table row cell negative 3 end cell row cell negative 3 end cell row 0 end table close parentheses end style

Misal begin mathsize 12px style Z with rightwards arrow on top end style adalah proyeksi vektor begin mathsize 12px style stack A C with rightwards arrow on top end style pada vektor begin mathsize 12px style stack A B with rightwards arrow on top end style, maka:

begin mathsize 12px style Z with rightwards arrow on top equals fraction numerator stack A C with rightwards arrow on top. 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  Z with rightwards arrow on top equals fraction numerator open parentheses table row cell negative 3 end cell row 0 row 4 end table close parentheses. open parentheses table row cell negative 3 end cell row cell negative 3 end cell row 0 end table close parentheses over denominator square root of open parentheses negative 3 close parentheses squared plus open parentheses negative 3 close parentheses squared plus 0 squared end root end fraction  Z with rightwards arrow on top equals fraction numerator 9 plus 0 plus 0 over denominator square root of 18 end fraction  Z with rightwards arrow on top equals fraction numerator 9 over denominator 3 square root of 2 end fraction  Z with rightwards arrow on top equals fraction numerator 3 over denominator square root of 2 end fraction  Z with rightwards arrow on top equals 3 over 2 square root of 2 end style

Jadi, panjang vektor begin mathsize 12px style stack A C with rightwards arrow on top end style ke begin mathsize 12px style stack A B with rightwards arrow on top end style adalah begin mathsize 12px style 3 over 2 square root of 2 end style

0

Roboguru

Diketahui vektor dan . Jika || adalah panjang proyeksi vektor pada, dan || = 4,maka nilai p adalah....

Pembahasan Soal:

open vertical bar straight c with rightwards arrow on top close vertical bar equals 4  open vertical bar fraction numerator straight a with rightwards arrow on top times straight b with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar end fraction close vertical bar equals 4  open vertical bar fraction numerator open parentheses table row 2 row cell negative straight p end cell row 3 end table close parentheses times open parentheses table row 1 row cell negative 2 end cell row 2 end table close parentheses over denominator square root of 1 squared plus open parentheses negative 2 close parentheses squared plus 2 squared end root end fraction close vertical bar equals 4  open vertical bar fraction numerator 2 plus 2 straight p plus 6 over denominator 3 end fraction close vertical bar equals 4  fraction numerator 2 plus 2 straight p plus 6 over denominator 3 end fraction equals 4 space atau space fraction numerator 2 plus 2 straight p plus 6 over denominator 3 end fraction equals negative 4  2 plus 2 straight p plus 6 equals 12 space space space space space space space space space space 2 plus 2 straight p plus 6 equals negative 12  2 straight p plus 8 equals 12 space space space space space space space space space space space space space space space space space 2 straight p plus 8 equals negative 12  2 straight p equals 4 space space space space space space space space space space space space space space space space space space space space space space space space space space 2 straight p equals negative 20  bold p bold equals bold 2 bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space space space space space space space space space space space space space space space straight p equals negative 10

0

Roboguru

Diketahui  maka panjang proyeksi  pada  adalah ...

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator a times b over denominator open vertical bar b close vertical bar end fraction end cell equals cell fraction numerator 2 times 1 plus 2 times left parenthesis negative 2 right parenthesis plus left parenthesis negative 4 right parenthesis times 2 over denominator square root of 1 squared plus left parenthesis negative 2 right parenthesis squared plus 2 to the power of 2 end exponent end root end fraction end cell row blank equals cell fraction numerator 2 plus open parentheses negative 4 close parentheses plus open parentheses negative 8 close parentheses over denominator square root of 9 end fraction end cell row blank equals cell negative 10 over 3 end cell end table end style
 

Panjang proyeksi begin mathsize 14px style a end style pada begin mathsize 14px style b end style adalah begin mathsize 14px style negative 10 over 3 end style 

1

Roboguru

Tentukan panjang proyeksi vektor  pada vektor .

Pembahasan Soal:

Ubahlah terlebih dahulu vektor basis ke dalam bentuk vektor kolom.

Vektor a with rightwards arrow on top equals i with rightwards arrow on top plus 2 j with rightwards arrow on top plus k with rightwards arrow on top, maka a with rightwards arrow on top equals open parentheses 1 comma space 2 comma space 1 close parentheses.

Vektor b with rightwards arrow on top equals 4 i with rightwards arrow on top minus 4 j with rightwards arrow on top plus 7 k with rightwards arrow on top, maka b with rightwards arrow on top equals open parentheses 4 comma space minus 4 comma space 7 close parentheses.

Rumus panjang proyeksi vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top yaitu:

open vertical bar p 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

Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar p 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 blank equals cell fraction numerator 1 open parentheses 4 close parentheses plus 2 open parentheses negative 4 close parentheses plus 1 open parentheses 7 close parentheses over denominator square root of 4 squared plus open parentheses negative 4 close parentheses squared plus 7 squared end root end fraction end cell row blank equals cell fraction numerator 4 plus open parentheses negative 8 close parentheses plus 7 over denominator square root of 16 plus 16 plus 49 end root end fraction end cell row blank equals cell fraction numerator 4 minus 8 plus 7 over denominator square root of 81 end fraction end cell row blank equals cell 3 over 9 end cell row blank equals cell 1 third end cell end table

Maka, panjang proyeksi vektor a with rightwards arrow on top equals i with rightwards arrow on top plus 2 j with rightwards arrow on top plus k with rightwards arrow on top pada vektor b with rightwards arrow on top equals 4 i with rightwards arrow on top minus 4 j with rightwards arrow on top plus 7 k with rightwards arrow on top adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 third end cell end table.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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