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Tentukan proyeksi ortogonal vektor  pada vektor

Pertanyaan

Tentukan proyeksi ortogonal vektor begin mathsize 14px style a with rightwards harpoon with barb upwards on top equals open parentheses table row 10 row cell negative 5 end cell end table close parentheses end style pada vektor begin mathsize 14px style b with rightwards harpoon with barb upwards on top equals open parentheses table row cell negative 1 end cell row 2 end table close parentheses end style 

Pembahasan Soal:

Perhatikan bahwa

1. begin mathsize 14px style a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top equals open parentheses 10 cross times open parentheses negative 1 close parentheses close parentheses plus open parentheses open parentheses negative 5 close parentheses cross times 2 close parentheses equals negative 10 plus open parentheses negative 10 close parentheses equals negative 20 end style 

2. begin mathsize 14px style open vertical bar b with rightwards harpoon with barb upwards on top close vertical bar squared equals open parentheses square root of open parentheses negative 1 close parentheses squared plus 2 squared end root close parentheses squared equals open parentheses square root of 1 plus 4 end root close parentheses squared equals 1 plus 4 equals 5 end style

Sehingga diperoleh proyeksi ortogonal vektor begin mathsize 14px style a with rightwards harpoon with barb upwards on top end style pada vektor begin mathsize 14px style b with rightwards harpoon with barb upwards on top end style adalah

begin mathsize 14px style fraction numerator a with rightwards harpoon with barb upwards on top times b with rightwards harpoon with barb upwards on top over denominator open vertical bar b with rightwards harpoon with barb upwards on top close vertical bar squared end fraction b with rightwards harpoon with barb upwards on top equals fraction numerator negative 20 over denominator 5 end fraction open parentheses table row cell negative 1 end cell row 2 end table close parentheses equals negative 4 open parentheses table row cell negative 1 end cell row 2 end table close parentheses equals open parentheses table row 4 row cell negative 8 end cell end table close parentheses end style 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

K. Prameswari

Mahasiswa/Alumni Universitas Gadjah Mada

Terakhir diupdate 29 Maret 2021

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Pertanyaan yang serupa

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

Pembahasan Soal:

vertical line c with rightwards arrow on top vertical line equals fraction numerator a with rightwards arrow on top. b with rightwards arrow on top over denominator vertical line b with rightwards arrow on top vertical line end fraction  fraction numerator 6 over denominator square root of 26 end fraction equals fraction numerator open parentheses table row 2 row cell negative 2 p end cell row 4 end table close parentheses. open parentheses table row 1 row cell negative 3 end cell row 4 end table close parentheses over denominator square root of 1 squared plus left parenthesis negative 3 right parenthesis squared plus 4 squared end root end fraction  fraction numerator 6 over denominator square root of 26 end fraction equals fraction numerator 2 plus 6 p plus 16 over denominator square root of 26 end fraction  Maka  6 equals 2 plus 6 p plus 16  6 p equals 6 minus 2 minus 16  6 p equals negative 12  p equals negative 2

0

Roboguru

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

Pembahasan Soal:

Diketahui a with rightwards arrow on top equals open parentheses table row p row cell negative 2 end cell row 2 end table close parenthesesb with rightwards arrow on top equals open parentheses table row 2 row q row 4 end table close parentheses, dan c with rightwards arrow on top equals open parentheses table row 1 row cell negative 3 end cell row r end table close parentheses dengan c with rightwards arrow on top adalah proyeksi vektor a with rightwards arrow on top pada b with rightwards arrow on top. Proyeksi vektor a with rightwards arrow on top pada b with rightwards arrow on top sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell open parentheses table row 1 row cell negative 3 end cell row r end table close parentheses end cell row cell fraction numerator open parentheses a with rightwards arrow on top times b with rightwards arrow on top close parentheses over denominator open vertical bar b with rightwards arrow on top close vertical bar squared end fraction b with rightwards arrow on top end cell equals cell open parentheses table row 1 row cell negative 3 end cell row r end table close parentheses end cell row cell fraction numerator open parentheses p times 2 plus open parentheses negative 2 close parentheses times q plus 2 times 4 close parentheses over denominator open parentheses square root of 2 squared plus q squared plus 4 squared end root close parentheses squared end fraction times open parentheses table row 2 row q row 4 end table close parentheses end cell equals cell open parentheses table row 1 row cell negative 3 end cell row r end table close parentheses end cell row cell fraction numerator open parentheses 2 p minus 2 q plus 8 close parentheses over denominator 4 plus q squared plus 16 end fraction. open parentheses table row 2 row q row 4 end table close parentheses end cell equals cell open parentheses table row 1 row cell negative 3 end cell row r end table close parentheses end cell row cell fraction numerator open parentheses 2 p minus 2 q plus 8 close parentheses over denominator 20 plus q squared end fraction times open parentheses table row 2 row q row 4 end table close parentheses end cell equals cell open parentheses table row 1 row cell negative 3 end cell row r end table close parentheses end cell row cell open parentheses table row cell fraction numerator 2 open parentheses 2 p minus 2 q plus 8 close parentheses over denominator 20 plus q squared end fraction end cell row cell fraction numerator q open parentheses 2 p minus 2 q plus 8 close parentheses over denominator 20 plus q squared end fraction end cell row cell fraction numerator 4 open parentheses 2 p minus 2 q plus 8 close parentheses over denominator 20 plus q squared end fraction end cell end table close parentheses end cell equals cell open parentheses table row 1 row cell negative 3 end cell row r end table close parentheses end cell end table

Berdasarkan sifat kesamaan matriks, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 2 open parentheses 2 p minus 2 q plus 8 close parentheses over denominator 20 plus q squared end fraction end cell equals 1 row cell 2 cross times open parentheses fraction numerator 2 p minus 2 q plus 8 over denominator 20 plus q squared end fraction close parentheses end cell equals 1 row cell fraction numerator 2 p minus 2 q plus 8 over denominator 20 plus q squared end fraction end cell equals cell 1 half space left parenthesis Pers space 1 right parenthesis end cell end table


table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator q open parentheses 2 p minus 2 q plus 8 close parentheses over denominator 20 plus q squared end fraction end cell equals cell negative 3 end cell row cell q cross times open parentheses fraction numerator 2 p minus 2 q plus 8 over denominator 20 plus q squared end fraction close parentheses end cell equals cell negative 3 end cell row cell q cross times 1 half end cell equals cell negative 3 end cell row q equals cell negative 3 cross times 2 end cell row blank equals cell negative 6 end cell end table
 

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 open parentheses 2 p plus 2 q plus 8 close parentheses over denominator 20 plus q squared end fraction end cell equals r row cell 4 cross times open parentheses fraction numerator 2 p minus 2 q plus 8 over denominator 20 plus q squared end fraction close parentheses end cell equals r row cell 4 cross times 1 half end cell equals r row r equals 2 row blank blank blank end table

Kemudian, substitusi q equals negative 6 pada persamaan 1 sehingga diperoleh nilai p sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 2 p minus 2 q plus 8 over denominator 20 plus q squared end fraction end cell equals cell 1 half end cell row cell fraction numerator 2 p minus 2 open parentheses negative 6 close parentheses plus 8 over denominator 20 plus open parentheses negative 6 close parentheses squared end fraction end cell equals cell 1 half end cell row cell fraction numerator 2 p plus 12 plus 8 over denominator 20 plus 36 end fraction end cell equals cell 1 half end cell row cell fraction numerator 2 p plus 20 over denominator 56 end fraction end cell equals cell 1 half end cell row cell 2 p plus 20 end cell equals cell 56 over 2 end cell row cell 2 p plus 20 end cell equals 28 row cell 2 p end cell equals cell 28 minus 20 end cell row blank equals 8 row p equals cell 8 over 2 end cell row blank equals 4 end table

Berdasarkan uraian di atas, diperoleh nilai p equals 4q equals negative 6, dan r equals 2, maka nilai p plus q plus r sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell p plus q plus r end cell equals cell 4 plus open parentheses negative 6 close parentheses plus 2 end cell row blank equals 0 end table


Oleh karena itu, jawaban yang benar adalah D.

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Diketahui vektor . Proyeksi vektor orthogonal pada vektor adalah ....

Pembahasan Soal:

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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 ⃗ p a d a space v ⃗ equals fraction numerator u with rightwards arrow on top. space v with rightwards arrow on top over denominator vertical line v ⃗ vertical line squared end fraction. v ⃗  equals fraction numerator 7. left parenthesis negative 2 right parenthesis plus left parenthesis negative 4 right parenthesis. left parenthesis negative 1 right parenthesis plus 1.0 over denominator left parenthesis negative 2 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared plus 0 squared end fraction open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 0 end table close parentheses  equals fraction numerator negative 10 over denominator 5 end fraction open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 0 end table close parentheses  equals negative 2 open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 0 end table close parentheses  equals open parentheses table row 4 row 2 row 0 end table close parentheses

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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

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