Roboguru

Jika ∣a∣=2,∣∣​b∣∣​=4dan∠(a,b)=30∘ maka proyeksi skalar orthogonal a pada b =...

Pertanyaan

Jika open vertical bar top enclose a close vertical bar equals 2 comma space open vertical bar top enclose b close vertical bar equals 4 space dan space angle left parenthesis top enclose a comma space top enclose b right parenthesis equals 30 degree maka proyeksi skalar orthogonal top enclose a pada top enclose b =...

  1. space 4 square root of 2   

  2. space 4 square root of 3 space 

  3. space 2 square root of 3   

  4. space square root of 3   

  5. 1 

Pembahasan Soal:

Ingat kembali proyeksi skalar orthogonal berikut.

Jika vektor top enclose a diproyeksikan secara orthogonal pada top enclose b, maka proyeksi skalar orthogonalnya adalah 

open vertical bar top enclose c close vertical bar equals open vertical bar open vertical bar top enclose a close vertical bar space cos space alpha close vertical bar 

Dari rumus di atas, diperoleh perhitungan sebagai berikut.

Proyeksi skalar orthogonal vektor top enclose a pada top enclose b adalah

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 open vertical bar top enclose a close vertical bar space cos space alpha close vertical bar end cell row blank equals cell vertical line 2 space cos space 30 degree vertical line end cell row blank equals cell vertical line 2 space left parenthesis 1 half square root of 3 right parenthesis vertical line end cell row blank equals cell vertical line square root of 3 vertical line end cell row blank equals cell square root of 3 end cell row blank blank blank row blank blank blank end table   

Jadi, proyeksi skalar orthogonal vektor top enclose a pada top enclose b adalah square root of 3.

Oleh karena itu, jawaban yang benar adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Jika vektor a tegak lurus b maka proyeksi skalar orthogonal   pada  sama dengan...

Pembahasan Soal:

Ingat kembali proyeksi skalar orthogonal berikut.

Jika vektor top enclose a diproyeksikan secara orthogonal pada top enclose b, maka proyeksi skalar orthogonalnya adalah 

open vertical bar top enclose c close vertical bar equals open vertical bar open vertical bar top enclose a close vertical bar space cos space alpha close vertical bar 

Dari rumus di atas, diperoleh perhitungan sebagai berikut.

Vektor top enclose a tegak lurus top enclose b artinya angle left parenthesis top enclose a comma space top enclose b right parenthesis equals 90 degree.

Sehingga proyeksi skalar orthogonal vektor top enclose a pada top enclose b adalah

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 open vertical bar top enclose a close vertical bar space cos space alpha close vertical bar end cell row blank equals cell vertical line open vertical bar top enclose a close vertical bar space cos space 90 degree vertical line end cell row blank equals cell vertical line open vertical bar top enclose a close vertical bar space 0 vertical line end cell row blank equals cell vertical line 0 vertical line end cell row blank equals 0 row blank blank blank row blank blank blank end table  

Jadi, proyeksi skalar orthogonal vektor top enclose a pada top enclose b adalah 0.

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Diketahui vektor a⇀=⎝⎛​−234​⎠⎞​danb⇀=⎝⎛​x03​⎠⎞​. Jika panjang proyeksi vektor a pada b adalah 54​, maka salah satu nilai x adalah ....

Pembahasan Soal:

V e k t o r space a equals open parentheses table row cell negative 2 end cell row 3 row 4 end table close parentheses space d a n space b equals space open parentheses table row x row 0 row 3 end table close parentheses space P a n j a n g space p r o y e k s i space v e k t o r stack space a with rightwards arrow on top space p a d a space b with rightwards arrow on top space a d a l a h space 4 over 5  fraction numerator a with rightwards arrow on top. b with rightwards arrow on top over denominator stack open vertical bar b close vertical bar with rightwards arrow on top end fraction equals 4 over 5  fraction numerator negative 2 x plus 10 plus 12 over denominator square root of x squared plus 0 plus 32 end root end fraction equals 4 over 5  fraction numerator negative 2 x plus 12 over denominator square root of x squared plus 9 end root end fraction equals 4 over 5  5 left parenthesis negative 2 x space plus space 12 right parenthesis space equals space 4 square root of x squared plus 9 end root  left parenthesis negative 10 x plus 60 right parenthesis space equals space 4 square root of x squared plus 9 end root  left parenthesis negative 10 x plus 60 right parenthesis squared space equals space left parenthesis 42 space right parenthesis left parenthesis x 2 space plus 9 right parenthesis squared  100 x squared space minus 1200 x space plus space 3600 space equals space 16 space left parenthesis x squared space plus space 9 right parenthesis  100 x squared space minus 1200 x space plus space 3600 space equals space 16 x squared space plus space 144  84 x squared space minus 1200 x space plus space 3456 space equals space 0  7 space x squared space minus space 100 x space plus space 288 space equals space 0  left parenthesis 7 x minus 72 right parenthesis left parenthesis x minus 4 right parenthesis equals 0  x space equals space 72 over 7 space a t a u space x space equals space 4

0

Roboguru

Diketahui vektor u=5i+2j−8k dan v=3i−2j+6k. Tentukan a. Proyeksi skalar ortogonal 2u pada v

Pembahasan Soal:

Ingat kembali rumus proyeksi skalar ortogonal u pada v with rightwards arrow on top.

vuv 

Menentukan vektor 2 u with rightwards arrow on top terlebih dahulu.

2u==252810416

Sehingga, proyeksi skalar ortogonal 2 u with rightwards arrow on top pada v with rightwards arrow on top

v2uv====32+(2)2+(6)2[10416][326]9+4+3630896‬49‬74774 

Jadi, proyeksi skalar ortogonal 2 u with rightwards arrow on top pada v with rightwards arrow on topadalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 74 over 7 end cell end table.

0

Roboguru

Diketahui vektor u=⎝⎛​3t1​⎠⎞​ dan v=⎝⎛​352​⎠⎞​. Jika proyeksi skalar orthogonal vektor u pada vektor v=1545​​, maka t= ...

Pembahasan Soal:

Diketahui:

u with rightwards arrow on top equals open parentheses table row 3 row t row 1 end table close parentheses
v with rightwards arrow on top equals open parentheses table row 3 row 5 row 2 end table close parentheses
v with rightwards arrow on top equals fraction numerator 4 square root of 5 over denominator 15 end fraction

Pekalian vektor:

table attributes columnalign right center left columnspacing 0px end attributes row cell u with rightwards arrow on top times v with rightwards arrow on top end cell equals cell open parentheses table row 3 row t row 1 end table close parentheses times open parentheses table row 3 row 5 row 2 end table close parentheses end cell row blank equals cell 3 times 3 plus t times 5 plus 1 times 2 end cell row blank equals cell 9 plus 5 t plus 2 end cell row blank equals cell 11 plus 5 t end cell end table 

Panjang vektor v with rightwards arrow on top:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar v with rightwards arrow on top close vertical bar end cell equals cell square root of 3 squared plus 5 squared plus 2 squared end root end cell row blank equals cell square root of 9 plus 25 plus 4 end root end cell row blank equals cell square root of 38 end cell end table

Proyeksi orthogonal u with rightwards arrow on top pada v with rightwards arrow on top:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction end cell equals cell fraction numerator 4 square root of 5 over denominator 15 end fraction end cell row cell fraction numerator 11 plus 5 t over denominator square root of 38 end fraction end cell equals cell fraction numerator 4 square root of 5 over denominator 15 end fraction end cell row cell 15 open parentheses 11 plus 5 t close parentheses end cell equals cell 4 square root of 5 cross times square root of 38 end cell row cell 165 plus 75 t end cell equals cell 4 square root of 190 end cell row cell 75 t end cell equals cell 4 square root of 190 minus 165 end cell row t equals cell fraction numerator 4 square root of 190 minus 165 over denominator 75 end fraction end cell end table 

Sehingga, nilai t equals fraction numerator 4 square root of 190 minus 165 over denominator 75 end fraction.

Jadi, tidak ada pilihan jawaban yang tepat.

0

Roboguru

Diketahui vektor a=⎝⎛​34−2​⎠⎞​ dan b=⎝⎛​x2−1​⎠⎞​. Jika panjang proyeksi skalar ortogonal vektor a dan b adalah 34​, salah satu nilai x adalah ...

Pembahasan Soal:

Diketahui:

a with rightwards arrow on top equals open parentheses table row 3 row 4 row cell negative 2 end cell end table close parentheses
b with rightwards arrow on top equals open parentheses table row x row 2 row cell negative 1 end cell end table close parentheses
open double vertical bar a with rightwards arrow on top subscript b with rightwards arrow on top end subscript close double vertical bar equals 4 over 3 

Karena panjang proyeksi skalar ortogonal vektor a with rightwards arrow on top dan b with rightwards arrow on top adalah 4 over 3, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell open double vertical bar a with rightwards arrow on top subscript b with rightwards arrow on top end subscript close double vertical bar end cell equals cell 4 over 3 end cell row 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 equals cell 4 over 3 end cell row cell fraction numerator open parentheses table row 3 row 4 row cell negative 2 end cell end table close parentheses times open parentheses table row x row 2 row cell negative 1 end cell end table close parentheses over denominator square root of x squared plus 2 squared plus open parentheses negative 1 close parentheses squared end root end fraction end cell equals cell 4 over 3 end cell row cell fraction numerator 3 x plus 8 plus 2 over denominator square root of x squared plus 4 plus 1 end root end fraction end cell equals cell 4 over 3 end cell row cell fraction numerator 3 x plus 10 over denominator square root of x squared plus 5 end root end fraction end cell equals cell 4 over 3 end cell row cell 4 square root of x squared plus 5 end root end cell equals cell 3 open parentheses 3 x plus 10 close parentheses end cell row cell 4 square root of x squared plus 5 end root end cell equals cell 9 x plus 30 end cell row cell open parentheses 4 square root of x squared plus 5 end root close parentheses squared end cell equals cell open parentheses 9 x plus 30 close parentheses squared end cell row cell 16 open parentheses x squared plus 5 close parentheses end cell equals cell 81 x squared plus 540 x plus 900 end cell row cell 16 x squared plus 80 end cell equals cell 81 x squared plus 540 x plus 900 end cell row 0 equals cell 81 x squared minus 16 x squared plus 540 x plus 900 minus 80 end cell row 0 equals cell 65 x squared plus 540 x plus 820 end cell row 0 equals cell 13 x squared plus 108 x plus 164 end cell row 0 equals cell open parentheses x plus 2 close parentheses open parentheses 13 x plus 82 close parentheses end cell row x equals cell negative 2 space logical or space x equals negative 82 over 13 end cell end table

Sehingga, salah satu nilai xnya adalah negative 2.

Jadi, jawaban yang tepat adalah A.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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