Roboguru

Deketahui segitiga ABCD dengan koordinat A=(2,−1,−1), B(−1,4,−2) dan C(5,0,−3). Proyeksi vektor orthognal AB pada AC adalah ...

Pertanyaan

Deketahui segitiga ABCD dengan koordinat straight A equals open parentheses 2 comma space minus 1 comma space minus 1 close parenthesesstraight B open parentheses negative 1 comma space 4 comma space minus 2 close parentheses dan straight C open parentheses 5 comma space 0 comma space minus 3 close parentheses. Proyeksi vektor orthognal AB with rightwards arrow on top pada AC with rightwards arrow on top adalah ...

Pembahasan Soal:

Ingat konsep proyeksi vektor orthognal AB with rightwards arrow on top pada AC with rightwards arrow on top adalah 

Proyeksi space vektor space AC with rightwards arrow on top equals fraction numerator stack AB times with rightwards arrow on top AC with rightwards arrow on top over denominator open vertical bar AC with rightwards arrow on top close vertical bar squared end fraction stack times AC with rightwards arrow on top  

Diketahui straight A equals open parentheses 2 comma space minus 1 comma space minus 1 close parenthesesstraight B open parentheses negative 1 comma space 4 comma space minus 2 close parentheses dan straight C open parentheses 5 comma space 0 comma space minus 3 close parentheses maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell AB with rightwards arrow on top end cell equals cell straight B with rightwards arrow on top minus straight A with rightwards arrow on top end cell row blank equals cell open parentheses negative 1 comma space 4 comma space minus 2 close parentheses minus open parentheses 2 comma space minus 1 comma space minus 1 close parentheses end cell row blank equals cell open parentheses negative 3 comma space 5 comma space minus 1 close parentheses end cell row cell AC with rightwards arrow on top end cell equals cell straight C with rightwards arrow on top minus straight A with rightwards arrow on top end cell row blank equals cell open parentheses 5 comma space 0 comma space minus 3 close parentheses minus open parentheses 2 comma space minus 1 comma space minus 1 close parentheses end cell row blank equals cell open parentheses 3 comma space 1 comma space minus 2 close parentheses end cell end table

maka 

table attributes columnalign right center left columnspacing 0px end attributes row cell stack AB times with rightwards arrow on top AC with rightwards arrow on top end cell equals cell open parentheses negative 3 comma space 5 comma space minus 1 close parentheses times open parentheses 3 comma space 1 comma space minus 2 close parentheses end cell row blank equals cell negative 9 plus 5 plus 2 end cell row blank equals cell negative 2 end cell row cell open vertical bar AC with rightwards arrow on top close vertical bar end cell equals cell square root of 3 squared plus 1 squared plus open parentheses negative 2 close parentheses squared end root end cell row blank equals cell square root of 9 plus 1 plus 4 end root end cell row blank equals cell square root of 14 end cell end table 

sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell Proyeksi space vektor space AC with rightwards arrow on top end cell equals cell fraction numerator AB with rightwards arrow on top times AC with rightwards arrow on top over denominator open vertical bar AC with rightwards arrow on top close vertical bar squared end fraction times AC with rightwards arrow on top end cell row blank equals cell fraction numerator negative 2 over denominator open parentheses square root of 14 close parentheses squared end fraction open parentheses 3 comma space 1 comma space minus 2 close parentheses end cell row blank equals cell fraction numerator negative 2 over denominator 14 end fraction open parentheses 3 comma space 1 comma space minus 2 close parentheses end cell row blank equals cell negative 1 over 7 open parentheses 3 comma space 1 comma space minus 2 close parentheses end cell row blank equals cell open parentheses negative 3 over 7 comma space minus 1 over 7 comma space 2 over 7 close parentheses end cell end table  

Dengan demikian proyeksi vektor orthognal AB with rightwards arrow on top pada AC with rightwards arrow on top adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative 3 over 7 comma space minus 1 over 7 comma space 2 over 7 close parentheses end cell end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Amamah

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui vektor a=⎝⎛​−4−22​⎠⎞​ dan b=⎝⎛​5−14​⎠⎞​. Proyeksi orthogonal vektor a pada vektor b adalah ...

Pembahasan Soal:

Ingat konsep perkalian titik dari dua vektor dan panjang vektor, maka proyeksi orthogonal vektor straight a with rightwards arrow on top pada vektor straight b with rightwards arrow on top adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell straight c with rightwards arrow on top end cell equals cell 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 squared end fraction times straight b with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row cell negative 4 end cell row cell negative 2 end cell row 2 end table close parentheses times open parentheses table row 5 row cell negative 1 end cell row 4 end table close parentheses over denominator open parentheses square root of 5 squared plus open parentheses negative 1 close parentheses squared plus open parentheses 4 close parentheses squared end root close parentheses squared end fraction times open parentheses table row 5 row cell negative 1 end cell row 4 end table close parentheses end cell row blank equals cell fraction numerator open parentheses negative 4 cross times 5 close parentheses plus open parentheses negative 2 cross times negative 1 close parentheses plus open parentheses 2 cross times 4 close parentheses over denominator open parentheses square root of 25 plus 1 plus 16 end root close parentheses squared end fraction times open parentheses table row 5 row cell negative 1 end cell row 4 end table close parentheses end cell row blank equals cell fraction numerator negative 20 plus 2 plus 8 over denominator open parentheses square root of 42 close parentheses squared end fraction times open parentheses table row 5 row cell negative 1 end cell row 4 end table close parentheses end cell row blank equals cell fraction numerator negative 10 over denominator 42 end fraction times open parentheses table row 5 row cell negative 1 end cell row 4 end table close parentheses end cell row blank equals cell negative 5 over 21 times open parentheses table row 5 row cell negative 1 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 25 over 21 end cell row cell 5 over 21 end cell row cell negative 20 over 21 end cell end table close parentheses end cell end table 

Dengan demikian proyeksi orthogonal vektor straight a with rightwards arrow on top pada vektor straight b with rightwards arrow on top adalah adalah straight c with rightwards arrow on top equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row cell negative 25 over 21 end cell row cell 5 over 21 end cell row cell negative 20 over 21 end cell end table close parentheses end cell end table.

0

Roboguru

Diketahui titik-titik A(2,1,−3), B(10,1,3) dan C(8,4,−1). Jika titik D merupakan proyeksi titik C pada garis AB maka panjang AD 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 titik-titik A(1,−3,2), B(1,3,10) dan C(4,−1,8). Jika D merupakan proyeksi titik C pada garis AB maka AD 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

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

Komponen vektor stack A C with rightwards arrow on top dan stack A B with rightwards arrow on top dapat ditentukan sebagai berikut.

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 4 row cell negative 1 end cell row 8 end table close parentheses minus open parentheses table row 1 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell open parentheses table row 3 row 2 row 6 end table close parentheses end cell end table

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 1 row 3 row 10 end table close parentheses minus open parentheses table row 1 row cell negative 3 end cell row 2 end table close parentheses end cell row blank equals cell open parentheses table row 0 row 6 row 8 end table close parentheses end cell end table

Proyeksi vektor stack A C with rightwards harpoon with barb upwards on top pada stack A B with rightwards arrow on top adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell stack A D with rightwards arrow on top end cell equals cell 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 squared end fraction times stack A B with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 3 row 2 row 6 end table close parentheses open parentheses table row 0 row 6 row 8 end table close parentheses over denominator open parentheses square root of 0 squared plus 6 squared plus 8 squared end root close parentheses squared end fraction times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell fraction numerator 3 times 0 plus 2 times 6 plus 6 times 8 over denominator open parentheses square root of 100 close parentheses squared end fraction times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell 60 over 100 times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell 3 over 5 times open parentheses table row 0 row 6 row 8 end table close parentheses end cell row blank equals cell times open parentheses table row 0 row cell 18 over 5 end cell row cell 24 over 5 end cell end table close parentheses end cell end table

Diperoleh vektor stack A D with rightwards arrow on top equals 18 over 5 j with rightwards arrow on top plus 24 over 5 k with rightwards arrow on top 

Oleh karena itu, tidak terdapat pilihan jawaban yang tepat.

0

Roboguru

Vektor yang merupakan proyeksi vektor x=3i+j​−k pada vektor y​=2i+5j​+k adalah ...

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.

0

Roboguru

Misal vektor a=4i+yj​ dan b=2i+2j​+k. Jika panjang proyeksi vektor a pada b 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

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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