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Vektor yang merupakan proyeksi vektor  pada vektor  adalah ...

Pertanyaan

Vektor yang merupakan proyeksi vektor stack text x end text with rightwards arrow on top equals 3 stack text i end text with rightwards arrow on top plus stack text j end text with rightwards arrow on top minus stack text k end text with rightwards arrow on top pada vektor stack text y end text with rightwards arrow on top equals 2 stack text i end text with rightwards arrow on top plus 5 stack text j end text with rightwards arrow on top plus stack text k end text with rightwards arrow on top adalah ...

  1. stack text i end text with rightwards arrow on top plus 5 over 2 stack text j end text with rightwards arrow on top plus 1 half stack text k end text with rightwards arrow on top 

  2. 2 over 3 stack text i end text with rightwards arrow on top plus 5 over 3 stack text j end text with rightwards arrow on top plus 1 third stack text k end text with rightwards arrow on top 

  3. 1 half stack text i end text with rightwards arrow on top plus 5 over 4 stack text j end text with rightwards arrow on top plus 1 fourth stack text k end text with rightwards arrow on top 

  4. 2 over 5 stack text i end text with rightwards arrow on top minus stack text j end text with rightwards arrow on top minus 1 fifth stack text k end text with rightwards arrow on top 

  5. 1 third stack text i end text with rightwards arrow on top plus 5 over 6 stack text j end text with rightwards arrow on top plus 1 over 6 stack text k end text with rightwards arrow on top 

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.

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

Deketahui segitiga ABCD dengan koordinat ,  dan . Proyeksi vektor orthognal  pada  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.

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Roboguru

Diketahui vektor  dan . Proyeksi orthogonal vektor  pada vektor  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 ,  dan . Jika  merupakan proyeksi titik  pada garis  maka  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.

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Roboguru

Diketahui vektor Proyeksi vektor orthogonal pada adalah...

Pembahasan Soal:

P r o y e k s i space v e k t o r space o r t h o g o n a l space a with rightwards arrow on top space p a d a space b with rightwards arrow on top comma space m i s a l k a n space c with rightwards arrow on top  c with ⃗ on top equals fraction numerator a with ⃗ on top straight space. b with ⃑ on top over denominator open vertical bar b with ⃑ on top close vertical bar squared end fraction. b with ⃑ on top equals fraction numerator open parentheses table row cell negative 1 end cell row cell negative 1 end cell row 2 end table close parentheses straight space. open parentheses table row 1 row cell negative 1 end cell row cell negative 2 end cell end table close parentheses over denominator open parentheses 1 close parentheses squared plus open parentheses negative 1 close parentheses squared plus open parentheses negative 2 close parentheses squared end fraction. open parentheses table row 1 row cell negative 1 end cell row cell negative 2 end cell end table close parentheses  equals fraction numerator open parentheses negative 1 close parentheses. open parentheses 1 close parentheses plus open parentheses negative 1 close parentheses. open parentheses negative 1 close parentheses plus 2 open parentheses negative 2 close parentheses over denominator 6 end fraction. open parentheses table row 1 row cell negative 1 end cell row cell negative 2 end cell end table close parentheses  equals fraction numerator negative 4 over denominator 6 end fraction open parentheses table row 1 row cell negative 1 end cell row cell negative 2 end cell end table close parentheses  c with ⃗ on top equals negative 2 over 3 open parentheses table row 1 row cell negative 1 end cell row cell negative 2 end cell end table close parentheses space a t a u space c with ⃗ on top equals negative 2 over 3 i plus 2 over 3 j plus 4 over 3 k

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Roboguru

Diketahui vektor Proyeksi vektor pada adalah....

Pembahasan Soal:

P r o y e k s i space v e k t o r space o r t h o g o n a l space u space ⃗ space p a d a space v space ⃑ comma space m i s a l k a n space c space ⃗  c with ⃗ on top equals fraction numerator u with ⃑ on top straight space. v with ⃑ on top over denominator open vertical bar v with ⃑ on top close vertical bar squared end fraction. v with ⃑ on top equals fraction numerator open parentheses table row cell negative 4 end cell row 4 row 3 end table close parentheses straight space. open parentheses table row cell negative 3 end cell row cell negative 6 end cell row 0 end table close parentheses over denominator open parentheses negative 3 close parentheses squared plus open parentheses negative 6 close parentheses squared plus open parentheses 0 close parentheses squared end fraction. open parentheses table row cell negative 3 end cell row cell negative 6 end cell row 0 end table close parentheses  equals fraction numerator open parentheses negative 4 close parentheses. open parentheses negative 3 close parentheses plus open parentheses 4 close parentheses. open parentheses negative 6 close parentheses plus 3 open parentheses 0 close parentheses over denominator 45 end fraction. open parentheses table row cell negative 3 end cell row cell negative 6 end cell row 0 end table close parentheses  equals fraction numerator negative 12 over denominator 45 end fraction open parentheses table row cell negative 3 end cell row cell negative 6 end cell row 0 end table close parentheses  c with ⃗ on top equals negative 4 over 15 open parentheses table row cell negative 3 end cell row cell negative 6 end cell row 0 end table close parentheses equals open parentheses table row cell 4 over 5 end cell row cell 8 over 5 end cell row 0 end table close parentheses space a t a u space space c with ⃗ on top equals 4 over 5 i plus 8 over 5 j plus 0 k

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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