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

Pertanyaan

Diketahui titik-titik text A end text open parentheses 2 comma space 1 comma space minus 3 close parenthesestext B end text open parentheses 10 comma space 1 comma space 3 close parentheses dan text C end text open parentheses 8 comma space 4 comma space minus 1 close parentheses. Jika titik text D end text merupakan proyeksi titik straight C pada garis text AB end text maka panjang text AD end text sama dengan ...

  1. 4 

  2. 5 

  3. 6 

  4. 7 

  5. 8 

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.

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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Pertanyaan yang serupa

Diketahui vektor  dan vektor . Panjang proyeksi vektor  pada vektor  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

Berdasarkan konsep di atas, 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 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 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

Oleh karena itu, jawaban yang tepat adalah C.

0

Roboguru

Diketahui titik . Panjang proyeksi vektor  pada  sama dengan...

Pembahasan Soal:

Ingat kembali panjang proyeksi berikut.

  • Jika vektor top enclose a diproyeksikan secara orthogonal pada top enclose b, maka panjang proyeksi vektornya 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 AB  

table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose AB end cell equals cell straight B minus straight A end cell row blank equals cell left parenthesis 5 comma space 2 comma space 7 right parenthesis minus left parenthesis 3 comma space 2 comma space 4 right parenthesis end cell row blank equals cell left parenthesis 5 minus 3 comma space 2 minus 2 comma space 7 minus 4 right parenthesis end cell row blank equals cell left parenthesis 2 comma space 0 comma space 3 right parenthesis end cell end table   

Menentukan top enclose AC   

table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose A C end enclose end cell equals cell C minus straight A end cell row blank equals cell left parenthesis 2 comma space 0 comma space 6 right parenthesis minus left parenthesis 3 comma space 2 comma space 4 right parenthesis end cell row blank equals cell left parenthesis 2 minus 3 comma space 0 minus 2 comma space 6 minus 4 right parenthesis end cell row blank equals cell left parenthesis negative 1 comma space minus 2 comma space 2 right parenthesis end cell end table    

Panjang proyeksi  top enclose AB pada top enclose AC adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose d end cell equals cell open vertical bar fraction numerator top enclose AB bullet top enclose AC over denominator open vertical bar top enclose AC close vertical bar end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator left parenthesis 2 comma space 0 comma space 3 right parenthesis bullet left parenthesis negative 1 comma space minus 2 comma space 2 right parenthesis over denominator square root of left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared plus 2 squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 2 left parenthesis negative 1 right parenthesis plus 0 left parenthesis negative 2 right parenthesis plus 3 left parenthesis 2 right parenthesis over denominator square root of 1 plus 4 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 2 plus 0 plus 6 over denominator square root of 9 end fraction close vertical bar end cell row blank equals cell open vertical bar 4 over 3 close vertical bar end cell row blank equals cell 4 over 3 end cell row blank equals cell 1 1 third end cell end table  

Jadi, panjang proyeksi  top enclose AB pada top enclose AC adalah 1 1 third.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Diketahui vektor  dan . Jika  adalah panjang proyeksi vektor  pada  dan , nilai  adalah ....

Pembahasan Soal:

Jika vektor c with rightwards arrow on top adalah proyeksi vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top, maka panjang proyeksi vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top yaitu open vertical bar c with rightwards arrow on top close vertical bar dapat ditentukan dengan 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 

Sehingga dapat ditentukan nilai p 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 4 equals cell fraction numerator open parentheses table row 2 row cell negative 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 end cell row blank equals cell fraction numerator 2 plus 2 p plus 6 over denominator square root of 9 end fraction end cell row 4 equals cell fraction numerator 2 p plus 8 over denominator 3 end fraction end cell row 12 equals cell 2 p plus 8 end cell row cell 2 p end cell equals cell 12 minus 8 end cell row p equals cell 4 over 2 end cell row p equals 2 end table 

Jadi, jawaban yang benar adalah D.

0

Roboguru

Misal vektor  dan . Jika panjang proyeksi vektor  pada  sama dengan 4, maka nilai y = ...

Pembahasan Soal:

Ingat kembali panjang proyeksi berikut.

  • Jika vektor top enclose a diproyeksikan secara orthogonal pada top enclose b, maka panjang proyeksi vektornya 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.

Karena  panjang proyeksi vektor top enclose a pada top enclose b sama dengan 4, maka

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 4 equals cell open vertical bar fraction numerator 4 left parenthesis 2 right parenthesis plus y left parenthesis 2 right parenthesis plus 0 left parenthesis 1 right parenthesis over denominator square root of 2 squared plus 2 squared plus 1 squared end root end fraction close vertical bar end cell row 4 equals cell open vertical bar fraction numerator 8 plus 2 y over denominator square root of 4 plus 4 plus 1 end root end fraction close vertical bar end cell row 4 equals cell open vertical bar fraction numerator 8 plus 2 y over denominator square root of 9 end fraction close vertical bar end cell row 4 equals cell open vertical bar fraction numerator 8 plus 2 y over denominator 3 end fraction close vertical bar end cell row 4 equals cell fraction numerator 8 plus 2 y over denominator 3 end fraction end cell row 12 equals cell 8 plus 2 y end cell row cell 12 minus 8 end cell equals cell 2 y end cell row 4 equals cell 2 y end cell row cell 4 over 2 end cell equals y row 2 equals y row y equals 2 end table 

Jadi nilai y adalah 2.  

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Misal vektor  dan vektor . Jika panjang proyeksi vektor  pada  sama dengan 3, maka nilai z yang tidak bulat sama dengan...

Pembahasan Soal:

Ingat kembali panjang proyeksi berikut.

  • Jika vektor top enclose a diproyeksikan secara orthogonal pada top enclose b, maka panjang proyeksi vektornya 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.

Karena  panjang proyeksi vektor top enclose a pada top enclose b sama dengan 3, maka

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 3 equals cell open vertical bar fraction numerator 1 left parenthesis 3 right parenthesis plus left parenthesis negative 3 right parenthesis left parenthesis negative 2 right parenthesis plus 2 left parenthesis z right parenthesis over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus z squared end root end fraction close vertical bar end cell row 3 equals cell open vertical bar fraction numerator 3 plus 6 plus 2 z over denominator square root of 9 plus 4 plus z squared end root end fraction close vertical bar end cell row 3 equals cell open vertical bar fraction numerator 9 plus 2 z over denominator square root of 13 plus z squared end root end fraction close vertical bar space space end cell row 9 equals cell fraction numerator 81 plus 36 z plus 4 z squared over denominator 13 plus z squared end fraction end cell row cell 9 left parenthesis 13 plus z squared right parenthesis end cell equals cell 81 plus 36 z plus 4 z squared end cell row cell 117 plus 9 z squared end cell equals cell 81 plus 36 z plus 4 z squared end cell row cell 9 z squared minus 4 z squared minus 36 z plus 117 minus 81 end cell equals 0 row cell 5 z squared minus 36 z plus 36 end cell equals 0 row cell left parenthesis 5 z minus 6 right parenthesis left parenthesis z minus 6 right parenthesis end cell equals 0 row z equals cell 6 over 5 space atau space z equals 6 end cell end table    

Karena yang diminta adalah nilai z yang tidak bulat, maka

z equals 6 over 5 equals 1 1 fifth  

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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