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Diketahuivektoea=i−2j+kdanvektorb=3i+j−2k.Vektorcmewakilivektorhasilproyeksiorthogonalvektorbpadavektora,makavektorc=...

Pertanyaan

D i k e t a h u i space v e k t o e space a with rightwards arrow on top equals i minus 2 j plus k space d a n space v e k t o r space b with rightwards arrow on top equals 3 i plus j minus 2 k. space V e k t o r space c space m e w a k i l i space v e k t o r space h a s i l space p r o y e k s i space o r t h o g o n a l space v e k t o r space b with rightwards arrow on top space p a d a space v e k t o r space a with rightwards arrow on top comma space m a k a space v e k t o r space c with rightwards arrow on top equals...

  1. negative 1 over 6 left parenthesis i minus 2 j plus k right parenthesis

  2. negative 1 over 6 left parenthesis 3 i minus 2 j plus 2 k right parenthesis

  3. negative 1 over 14 left parenthesis i minus 2 j plus k right parenthesis

  4. negative 1 over 14 left parenthesis 3 i plus j plus 2 k right parenthesis

  5. 1 over 6 left parenthesis i minus 2 j plus k right parenthesis

Pembahasan Soal:

a space ⃗ equals open parentheses table row 1 row cell negative 2 end cell row 1 end table close parentheses comma b space ⃗ equals open parentheses table row 3 row 1 row cell negative 2 end cell end table close parentheses  c space ⃗ space m e m e n u h i space h u b u n g a n comma  c space ⃗ equals open parentheses fraction numerator a space ⃗ space space. b space ⃗ over denominator vertical line a ⃗ vertical line squared end fraction close parentheses a space ⃗  c space ⃗ equals open parentheses fraction numerator open parentheses table row 1 row cell negative 2 end cell row 1 end table close parentheses. open parentheses table row 3 row 1 row cell negative 2 end cell end table close parentheses over denominator open parentheses square root of 1 squared plus left parenthesis negative 2 right parenthesis squared plus 1 squared end root close parentheses squared end fraction close parentheses open parentheses table row 1 row cell negative 2 end cell row 1 end table close parentheses  c space ⃗ equals open parentheses fraction numerator 1 left parenthesis 3 right parenthesis minus 2 left parenthesis 1 right parenthesis plus 1 left parenthesis negative 2 right parenthesis over denominator 6 end fraction close parentheses open parentheses table row 1 row cell negative 2 end cell row 1 end table close parentheses  c space ⃗ equals negative 1 over 6 open parentheses table row 1 row cell negative 2 end cell row 1 end table close parentheses  c space ⃗ equals negative 1 over 6 left parenthesis i minus 2 j plus k right parenthesis

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Kurnia

Mahasiswa/Alumni Universitas Jember

Terakhir diupdate 04 Oktober 2021

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Pertanyaan yang serupa

Diketahui proyeksi skalar u→ pada v adalah −3. Jika  dan , maka vektor proyeksi u pada w adalah ....

Pembahasan Soal:

Perhatikan bahwa proyeksi skalar undefined pada undefined adalah begin mathsize 14px style negative sign 3 end style. Oleh karena itu, didapat

begin mathsize 14px style fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction equals minus sign 3 end style 

Perhatikan bahwa begin mathsize 14px style italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top end style pasti bernilai negatif karena begin mathsize 14px style italic v with rightwards arrow on top end style tidak mungkin bernilai negatif.

Jadi, sudut antara undefined dan undefined pasti membentuk sudut tumpul seperti digambarkan sebagai berikut.

Misalkan vektor proyeksi undefined pada undefined adalah begin mathsize 14px style italic p with rightwards arrow on top end style maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell italic p with rightwards arrow on top end cell equals cell open parentheses fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar squared end fraction close parentheses italic v with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar ⋅ open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses italic v with rightwards arrow on top end cell row blank equals cell open parentheses fraction numerator italic u with rightwards arrow on top ⋅ italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses open parentheses fraction numerator italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses end cell row blank equals cell negative sign 3 open parentheses fraction numerator italic v with rightwards arrow on top over denominator open vertical bar italic v with rightwards arrow on top close vertical bar end fraction close parentheses end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator open vertical bar v with rightwards arrow on top close vertical bar end fraction close parentheses italic v with rightwards arrow on top end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator square root of open parentheses negative sign 7 close parentheses squared plus open parentheses negative sign 4 close parentheses squared plus 4 squared end root end fraction close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator square root of 49 plus 16 plus 16 end root end fraction close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator square root of 81 end fraction close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign 3 over 9 close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses negative sign 1 third close parentheses open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell open parentheses table row cell 7 over 3 end cell row cell 4 over 3 end cell row cell negative sign 4 over 3 end cell end table close parentheses end cell row blank equals cell 7 over 3 italic i with rightwards arrow on top plus 4 over 3 italic j with rightwards arrow on top minus sign 4 over 3 italic k with rightwards arrow on top end cell end table end style 

Selanjutnya, dapat diperhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell italic w with rightwards arrow on top end cell equals cell open parentheses table row 14 row 8 row cell negative sign 8 end cell end table close parentheses end cell row blank equals cell negative sign 2 open parentheses table row cell negative sign 7 end cell row cell negative sign 4 end cell row 4 end table close parentheses end cell row blank equals cell negative sign 2 italic v with rightwards arrow on top end cell end table end style 

Jadi, vektor undefined dan undefined saling sejajar dengan arah yang berlawanan.

Perhatikan gambar berikut!

Dapat diperhatikan bahwa proyeksi vektor undefined pada undefined memiliki landasan yang sama dengan proyeksi vektor undefined pada undefined.

Jadi, proyeksi vektor undefined pada undefined sama saja dengan proyeksi vektor begin mathsize 14px style u with rightwards arrow on top end style pada undefined, yaitu vektor undefined.

Oleh karena itu, proyeksi vektor undefined pada undefined adalah

begin mathsize 14px style italic p with rightwards arrow on top equals 7 over 3 italic i with rightwards arrow on top plus 4 over 3 italic j with rightwards arrow on top minus sign 4 over 3 italic k with rightwards arrow on top end style 

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Berapa jumlah besaran? Sebutkan!

Pembahasan Soal:

Besaran dibagi menjadi 2, diantaranya:

  1. Besaran Skalar, yaitu besaran yang hanya mempunyai nilai.
  2. Besaran Vektor, yaitu besaran yang mempunyai nilai dan arah.

0

Roboguru

Diketahui vektor a=2i−2pj​+4k dan b=i−3j​+4k. Jika panjang proyeksi vektor a pada b adalah  26​6​, 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 vektor p​=5i−7j​+4k pada  q​=i−2j​+2k. Jika vektor r adalah proyeksi dari vektor p​ pada q​, maka = ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight r with rightwards arrow on top close vertical bar end cell equals cell fraction numerator straight p with rightwards arrow on top. straight q with rightwards arrow on top over denominator open vertical bar straight q with rightwards arrow on top close vertical bar squared end fraction straight q with rightwards arrow on top end cell row blank equals cell fraction numerator open parentheses 5 straight i with rightwards arrow on top minus 7 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top close parentheses space. space open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses over denominator open parentheses square root of 2 squared plus open parentheses negative 1 close parentheses squared plus open parentheses 2 close parentheses squared end root close parentheses squared end fraction open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell fraction numerator 5 plus 14 plus 8 over denominator open parentheses square root of 9 close parentheses squared end fraction open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell 27 over 9 open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell 3 open parentheses straight i with rightwards arrow on top minus 2 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses end cell row blank equals cell 3 straight i with rightwards arrow on top minus 6 straight j with rightwards arrow on top plus 6 straight k with rightwards arrow on top end cell end table end style 

1

Roboguru

Jika c merupakan vektor proyeksi a=2i−6j​+4k pada b=2i−j​+2k, maka panjang vektor  adalah ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar straight c with rightwards arrow on top close vertical bar end cell equals cell open vertical bar fraction numerator straight a with rightwards arrow on top. straight b with rightwards arrow on top over denominator open vertical bar straight 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 stack 2 straight i with rightwards arrow on top minus 6 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top close parentheses space. space open parentheses 2 straight i with rightwards arrow on top minus straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top close parentheses over denominator square root of 2 cubed plus open parentheses negative 1 close parentheses squared plus open parentheses 2 close parentheses squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 4 plus 6 plus 8 over denominator square root of 4 plus 1 plus 4 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 18 over denominator square root of 9 end fraction close vertical bar end cell row blank equals cell open vertical bar 18 over 3 close vertical bar end cell row blank equals 6 end table end style 

1

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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