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Diketahui vektor  dan vektor . Tentukan panjang proyeksi vektor a pada b dan panjang proyeksi vektor b pada a.

Pertanyaan

Diketahui vektor begin mathsize 14px style bold a equals open square brackets negative 2 comma 3 comma 4 close square brackets end style dan vektor begin mathsize 14px style bold b equals open square brackets 4 comma 0 comma 3 close square brackets end style. Tentukan panjang proyeksi vektor a pada b dan panjang proyeksi vektor b pada a.

  1. .... 

  2. .... 

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar bold b subscript bold a close vertical bar end cell equals cell fraction numerator open vertical bar bold a times bold b close vertical bar over denominator open vertical bar bold a close vertical bar end fraction end cell row blank equals cell fraction numerator open vertical bar negative 2 times 4 plus 3 times 0 plus 4 times 3 close vertical bar over denominator square root of open parentheses negative 2 close parentheses squared plus 3 squared plus 4 squared end root end fraction end cell row blank equals cell fraction numerator 4 over denominator square root of 29 end fraction end cell row blank equals cell 4 over 29 square root of 29 end cell end table end style

Jadi, panjang proyeksi b pada a adalah begin mathsize 14px style 4 over 29 square root of 29 end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Terakhir diupdate 29 Maret 2021

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Pertanyaan yang serupa

Diketahui koordinat . Jika AB wakil vektor dan AC wakil vektor , maka proyeksi pada adalah...

Pembahasan Soal:

Diketahui

koordinat A left parenthesis negative 4 comma 2 comma 3 right parenthesis comma space B left parenthesis 7 comma 8 comma negative 1 right parenthesis space d a n space C left parenthesis 1 comma 0 comma 7 right parenthesis. Jika AB wakil vektor u with rightwards arrow on top dan AC wakil vektor v with rightwards arrow on top ,

Ditanya :

maka proyeksi u with rightwards arrow on top pada v with rightwards arrow on top adalah...

Penyelesaian

table attributes columnalign right center left columnspacing 0px end attributes row cell u with rightwards harpoon with barb upwards on top end cell equals cell A B space end cell row cell u with rightwards arrow on top end cell equals cell space left parenthesis 7 comma 8 comma negative 1 right parenthesis space minus space left parenthesis negative 4 comma 2 comma 3 right parenthesis end cell row cell stack space u with rightwards arrow on top space end cell equals cell space left parenthesis 11 comma 6 comma negative 4 right parenthesis space space end cell row cell v with rightwards arrow on top space end cell equals cell space A C space end cell row cell v with rightwards arrow on top space end cell equals cell space left parenthesis 1 comma 0 comma 7 right parenthesis minus left parenthesis negative 4 comma 2 comma 3 right parenthesis space end cell row cell v with rightwards arrow on top space end cell equals cell space left parenthesis 5 comma negative 2 comma 4 right parenthesis end cell end table


Proyeksi u pada v adalah:.


table attributes columnalign right center left columnspacing 0px end attributes row cell straight u subscript straight v end cell equals cell fraction numerator straight u. straight v over denominator vertical line straight v vertical line ² end fraction. straight v end cell row cell straight u subscript straight v end cell equals cell fraction numerator left parenthesis 11.5 plus 6 left parenthesis negative 2 right parenthesis plus left parenthesis negative 4 right parenthesis 4 right parenthesis over denominator left parenthesis 5 ² plus left parenthesis negative 2 right parenthesis ² plus 4 ² right parenthesis end fraction left parenthesis 5 comma negative 2 comma 4 right parenthesis end cell row cell straight u subscript straight v end cell equals cell fraction numerator left parenthesis 55 minus 12 minus 16 right parenthesis over denominator left parenthesis 25 plus 4 plus 16 right parenthesis end fraction left parenthesis 5 comma negative 2 comma 4 right parenthesis end cell row cell straight u subscript straight v end cell equals cell 27 over 45 left parenthesis 5 comma negative 2 comma 4 right parenthesis end cell row cell straight u subscript straight v end cell equals cell 3 over 5 left parenthesis 5 comma negative 2 comma 4 right parenthesis end cell row cell straight u subscript straight v end cell equals cell left parenthesis 3 comma negative 6 over 5 comma 12 over 5 right parenthesis end cell row cell straight u subscript straight v end cell equals cell left parenthesis 3 straight i comma negative 6 over 5 straight j comma 12 over 5 straight k right parenthesis end cell end table

Jadi, royeksi u with rightwards arrow on top pada v with rightwards arrow on top adalah.3 i with rightwards arrow on top minus 6 over 5 j with rightwards arrow on top plus 12 over 5 k with rightwards arrow on top.

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

Diketahui  dan . Vektor proyeksi ortogonal  pada  adalah adalah ....

Pembahasan Soal:

Perhatikan bahwa

undefined  
 

Misalkan vektor proyeksi ortogonal undefined pada undefined adalah begin mathsize 14px style p with rightwards arrow on top end style. Oleh karena itu, didapatkan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator open parentheses u with rightwards arrow on top plus v with rightwards arrow on top close parentheses ⋅ v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar squared end fraction close parentheses v with rightwards arrow on top end cell equals cell open parentheses fraction numerator open parentheses table row 0 row cell negative sign 9 end cell row 0 end table close parentheses ⋅ open parentheses table row 2 row cell negative sign 1 end cell row cell negative sign 2 end cell end table close parentheses over denominator open parentheses square root of 2 squared plus open parentheses negative sign 1 close parentheses squared plus open parentheses negative sign 2 close parentheses squared end root close parentheses squared end fraction close parentheses open parentheses table row 2 row cell negative sign 1 end cell row cell negative sign 2 end cell end table close parentheses end cell row blank equals cell open parentheses fraction numerator 0 ⋅ 2 plus open parentheses negative sign 9 close parentheses ⋅ open parentheses negative sign 1 close parentheses plus 0 ⋅ open parentheses negative sign 2 close parentheses over denominator open parentheses square root of 4 plus 1 plus 4 end root close parentheses squared end fraction close parentheses open parentheses table row 2 row cell negative sign 1 end cell row cell negative sign 2 end cell end table close parentheses end cell row blank equals cell open parentheses fraction numerator 0 plus 9 plus 0 over denominator open parentheses square root of 9 close parentheses squared end fraction close parentheses open parentheses table row 2 row cell negative sign 1 end cell row cell negative sign 2 end cell end table close parentheses end cell row blank equals cell 9 over 9 open parentheses table row 2 row cell negative sign 1 end cell row cell negative sign 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row 2 row cell negative sign 1 end cell row cell negative sign 2 end cell end table close parentheses end cell row blank equals cell 2 i with rightwards arrow on top minus sign j with rightwards arrow on top minus sign 2 k with rightwards arrow on top end cell end table end style   
 

Jadi, jawaban yang tepat adalah C.

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Roboguru

Misal . Jika vektor  diproyeksikan pada vektor  maka panjang vektor proyeksinya = ...

Pembahasan Soal:

Perhatikan proyeksi vektor  top enclose a pada top enclose b berikut.

Misal sudut yang dibentuk oleh vektor top enclose a dan top enclose b adalah sudut lancip, dan proyeksi skalar orthogonalnya adalah vertical line top enclose c vertical line, maka

cos space alpha equals fraction numerator open vertical bar top enclose c close vertical bar over denominator open vertical bar top enclose a close vertical bar end fraction space atau space open vertical bar top enclose c close vertical bar equals open vertical bar top enclose a close vertical bar space cos space alpha 

Jika sudut yang dibentuk oleh   vektor top enclose a dan top enclose b adalah sudut tumpul dan proyeksi skalar orthogonalnyaadalah vertical line top enclose c vertical line, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator open vertical bar top enclose c close vertical bar over denominator open vertical bar top enclose a close vertical bar end fraction space end cell end table

atau 

table attributes columnalign right center left columnspacing 0px end attributes row cell space open vertical bar top enclose c close vertical bar end cell equals cell open vertical bar top enclose a close vertical bar space cos space left parenthesis 180 degree minus alpha right parenthesis end cell row blank equals cell negative open vertical bar a with bar on top close vertical bar space cos space alpha end cell end table

Perhatikan bahwa jika sudut alpha lancip, maka cos space alpha greater than 0 dan jika alpha tumpul maka cos space alpha less than 0. Ini berarti bahwa open vertical bar c with bar on top close vertical bar akan selalu greater or equal than 0. Dengan demikian dapat kita simpulkan bahwa proyeksi skalar orthogonal dari vektor a with bar on top pada b with bar on top adalah open vertical bar open vertical bar a with bar on top close vertical bar space cos space alpha close vertical bar.

Jadi, panjang vektor top enclose a pada top enclose b adalah open vertical bar open vertical bar a with bar on top close vertical bar space cos space alpha close vertical bar.

Oleh karena itu, jawaban yang benar adalah C.

0

Roboguru

Jika vektor  dan . Panjang proyeksi vektor  pada vektor  adalah ...

Pembahasan Soal:

Perhatikan gambar berikut.



 

Proyeksi vektor ortogonal a with rightwards arrow on top pada b with rightwards arrow on top dapat dirumuskan:

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  

dengan:

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell proyeksi space vektor space ortogonal space a with rightwards arrow on top space pada space b with rightwards arrow on top end cell row cell open vertical bar b with rightwards arrow on top close vertical bar end cell equals cell panjang space vektor space b with rightwards arrow on top end cell end table  

Panjang vektor b with rightwards arrow on top dapat dirumuskan:

open vertical bar b with rightwards arrow on top close vertical bar equals square root of b subscript 1 squared plus b subscript 2 squared end root 

dengan:

table attributes columnalign right center left columnspacing 0px end attributes row cell b subscript 1 end cell equals cell elemen space vektor space b with rightwards arrow on top space pada space sumbu space x end cell row cell b subscript 2 end cell equals cell elemen space vektor space stack b space with rightwards arrow on top pada space sumbu space y end cell end table 

Diketahui vektor p with rightwards arrow on top equals 5 j with rightwards arrow on top plus 2 i with rightwards arrow on top dan q with rightwards arrow on top equals 4 i with rightwards arrow on top minus 3 j with rightwards arrow on top, maka tentukan hasil dari p with rightwards arrow on top plus q with rightwards arrow on top.

Panjang proyeksi vektor p with rightwards arrow on top plus q with rightwards arrow on top pada vektor q with rightwards arrow on top:

table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell fraction numerator open parentheses p with rightwards arrow on top plus q with rightwards arrow on top close parentheses times q with rightwards arrow on top over denominator open vertical bar q with rightwards arrow on top close vertical bar squared end fraction times q with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses table row 6 row 2 end table close parentheses times open parentheses table row 4 row cell negative 3 end cell end table close parentheses over denominator open parentheses square root of 4 squared plus open parentheses negative 3 close parentheses squared end root close parentheses squared end fraction times open parentheses table row 4 row cell negative 3 end cell end table close parentheses end cell row blank equals cell fraction numerator 6 times 4 plus 2 times open parentheses negative 3 close parentheses over denominator 5 squared end fraction times open parentheses table row 4 row cell negative 3 end cell end table close parentheses end cell row blank equals cell 18 over 25 times open parentheses table row 4 row cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 18 over 25 times 4 end cell row cell 18 over 25 times open parentheses negative 3 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 2 comma 88 end cell row cell negative 2 comma 16 end cell end table close parentheses end cell end table 

Maka, proyeksi vektor p with rightwards arrow on top plus q with rightwards arrow on top pada vektor q with rightwards arrow on top adalah c with rightwards arrow on top equals 2 comma 88 i with rightwards arrow on top minus 2 comma 16 j with rightwards arrow on top.

Tentukan panjang proyeksi vektor p with rightwards arrow on top plus q with rightwards arrow on top pada vektor q with rightwards arrow on top.

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 square root of x squared plus y squared end root end cell row blank equals cell square root of 2 comma 88 squared plus open parentheses negative 2 comma 16 close parentheses squared end root end cell row blank equals cell square root of 12 comma 96 end root end cell row blank equals cell 3 comma 6 end cell end table 

Sehingga, panjang proyeksi vektor p with rightwards arrow on top plus q with rightwards arrow on top pada vektor q with rightwards arrow on top adalah 3 comma 6.

Jadi, jawaban yang tepat adalah C.

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

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