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

Pertanyaan

Diketahui proyeksi skalar begin mathsize 14px style italic u with italic rightwards arrow on top end style pada begin mathsize 14px style italic v with rightwards arrow on top end style adalah begin mathsize 14px style negative sign 3 end style. Jika undefined dan begin mathsize 14px style italic w with rightwards arrow on top equals open parentheses table row 14 row 8 row cell negative sign 8 end cell end table close parentheses end style, maka vektor proyeksi begin mathsize 14px style italic u with rightwards arrow on top end style pada begin mathsize 14px style italic w with rightwards arrow on top end style adalah ....

  1. begin mathsize 14px style 7 italic i with rightwards arrow on top plus 4 italic j with rightwards arrow on top minus sign 4 italic k with rightwards arrow on top end style 

  2. begin mathsize 14px style 14 italic i with rightwards arrow on top plus 8 italic j with rightwards arrow on top minus sign 8 italic k with rightwards arrow on top end style 

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

  4. begin mathsize 14px style negative sign 14 over 3 italic i with rightwards arrow on top minus sign 8 over 3 italic j with rightwards arrow on top plus 8 over 3 italic k with rightwards arrow on top end style 

  5. begin mathsize 14px style 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 

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Rizky

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 05 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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