Roboguru

Jika vektor  dan  segaris, maka tentukan nilai  dan .

Pertanyaan

Jika vektor a with rightwards arrow on top equals 4 i with rightwards arrow on top minus 2 j with rightwards arrow on top plus x k with rightwards arrow on top dan b with rightwards arrow on top equals negative 6 i with rightwards arrow on top plus y j with rightwards arrow on top plus 12 k with rightwards arrow on top segaris, maka tentukan nilai x dan y.

Pembahasan Soal:

Gunakan konsep dua vektor segaris, maka berlaku.

a with rightwards arrow on top equals k times b with rightwards arrow on top

Diketahui:
Vektor a with rightwards arrow on top equals 4 i with rightwards arrow on top minus 2 j with rightwards arrow on top plus x k with rightwards arrow on top dan b with rightwards arrow on top equals negative 6 i with rightwards arrow on top plus y j with rightwards arrow on top plus 12 k with rightwards arrow on top segaris.
Akan ditentukan nilai x dan y.

table attributes columnalign right center left columnspacing 2px end attributes row cell a with rightwards arrow on top end cell equals cell k times b with rightwards arrow on top end cell row cell open parentheses table row 4 row cell negative 2 end cell row x end table close parentheses end cell equals cell k open parentheses table row cell negative 6 end cell row y row 12 end table close parentheses end cell row cell open parentheses table row 4 row cell negative 2 end cell row x end table close parentheses end cell equals cell open parentheses table row cell negative 6 k end cell row cell k y end cell row cell 12 k end cell end table close parentheses end cell end table

Dari kesamaan dua vektor di atas, diperoleh:

table attributes columnalign right center left columnspacing 2px end attributes row 4 equals cell negative 6 k end cell row k equals cell fraction numerator 4 over denominator negative 6 end fraction end cell row k equals cell negative 2 over 3 end cell end table

Substitusikan nilai k equals negative 2 over 3 pada baris kedua diperoleh:

table attributes columnalign right center left columnspacing 2px end attributes row cell negative 2 end cell equals cell k y end cell row cell negative 2 end cell equals cell negative 2 over 3 y end cell row y equals cell fraction numerator negative 2 over denominator negative begin display style 2 over 3 end style end fraction end cell row y equals cell 2 cross times 3 over 2 end cell row y equals 3 end table

Kemudian substitusikan nilai k equals negative 2 over 3 pada baris ketiga diperoleh:

table attributes columnalign right center left columnspacing 2px end attributes row x equals cell 12 k end cell row x equals cell 12 cross times open parentheses negative 2 over 3 close parentheses end cell row x equals cell negative 24 over 3 end cell row x equals cell negative 8 end cell end table 

Jadi, diperoleh nilai x equals negative 8 dan y equals 3.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

Y. Fathoni

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Terakhir diupdate 01 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui vektor . Tunjukkan bahwa titik KLM segaris (kolinier).

Pembahasan Soal:

Titik KLM dikatakan segaris (kolinier) apabila begin mathsize 14px style stack K L with rightwards arrow on top equals k times stack L M with rightwards arrow on top end style

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Dari hasil perhitungan di atas, diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 9 end cell equals cell k times 8 end cell row k equals cell negative 9 over 8 end cell end table end style danbegin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 4 end cell equals cell k times 6 end cell row k equals cell negative 4 over 6 equals negative 2 over 3 end cell end table end style 

Diperoleh nilai begin mathsize 14px style k end style yang berbeda.

Dengan demikian, titik KLM tidak segaris (kolinier).

0

Roboguru

Diberikan vektor-vektor  dan vektor . Jika  dan  sejajar maka  sama dengan ...

Pembahasan Soal:

Jika vektor p with rightwards arrow on top sejajar vektor q with rightwards arrow on top, maka p with rightwards arrow on top equals k q with rightwards arrow on top dengan k skalar, k element of R

Pada soal di atas, apabila vektor a with rightwards arrow on top dan b with rightwards arrow on top sejajar, maka dapat ditentukan hubungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell k b with rightwards arrow on top end cell row cell open parentheses table row cell 2 x end cell row cell negative 2 y end cell row y end table close parentheses end cell equals cell k open parentheses table row cell negative 3 end cell row y row cell negative 1 end cell end table close parentheses end cell end table

Diperoleh negative 2 y equals k y space left right double arrow space k equals negative 2

Nilai x dan y adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell negative 3 k end cell row cell 2 x end cell equals cell negative 3 open parentheses negative 2 close parentheses end cell row cell 2 x end cell equals 6 row x equals 3 end table

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative k end cell row y equals cell negative open parentheses negative 2 close parentheses end cell row y equals 2 end table

Vektor a with rightwards arrow on top equals open parentheses table row cell 2 x end cell row cell negative 2 y end cell row y end table close parentheses equals open parentheses table row 6 row cell negative 4 end cell row 2 end table close parentheses dan vektor b with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row y row cell negative 1 end cell end table close parentheses equals open parentheses table row cell negative 3 end cell row 2 row cell negative 1 end cell end table close parentheses

Dapat ditentukan hasil operasi hitung pada vektor berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top plus 2 b with rightwards arrow on top end cell equals cell open parentheses table row 6 row cell negative 4 end cell row 2 end table close parentheses plus 2 open parentheses table row cell negative 3 end cell row 2 row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row 6 row cell negative 4 end cell row 2 end table close parentheses plus open parentheses table row cell negative 6 end cell row 4 row cell negative 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row 0 row 0 row 0 end table close parentheses end cell end table

Diperoleh a with rightwards arrow on top plus 2 b with rightwards arrow on top equals 0 with rightwards arrow on top 

Oleh karena itu, jawaban yang tepat adalah A.

0

Roboguru

Diberikan vektor-vektor  dan vektor . Jika  dan  sejajar maka

Pembahasan Soal:

Jika vektor p with rightwards arrow on top sejajar vektor q with rightwards arrow on top, maka p with rightwards arrow on top equals k q with rightwards arrow on top dengan k skalar, k element of R

Pada soal di atas, apabila vektor a with rightwards arrow on top dan b with rightwards arrow on top sejajar, maka dapat ditentukan hubungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell k b with rightwards arrow on top end cell row cell open parentheses table row x row cell negative 3 end cell row 6 end table close parentheses end cell equals cell k open parentheses table row cell 1 minus y end cell row 3 row cell negative 1 minus x end cell end table close parentheses end cell end table

Diperoleh negative 3 equals 3 k space left right double arrow space k equals negative 1

Nilai x dan y adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row 6 equals cell k open parentheses negative 1 minus x close parentheses end cell row 6 equals cell negative 1 open parentheses negative 1 minus x close parentheses end cell row 6 equals cell 1 plus x end cell row x equals 5 end table

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell k open parentheses 1 minus y close parentheses end cell row 5 equals cell negative 1 open parentheses 1 minus y close parentheses end cell row 5 equals cell negative 1 plus y end cell row y equals 6 end table

Vektor a with rightwards arrow on top equals open parentheses table row x row cell negative 3 end cell row 6 end table close parentheses equals open parentheses table row 5 row cell negative 3 end cell row 6 end table close parentheses dan vektor b with rightwards arrow on top equals open parentheses table row cell 1 minus y end cell row 3 row cell negative open parentheses 1 plus x close parentheses end cell end table close parentheses equals open parentheses table row cell negative 5 end cell row 3 row cell negative 6 end cell end table close parentheses

Dapat ditentukan hasil operasi hitung pada vektor berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top plus 3 b with rightwards arrow on top end cell equals cell open parentheses table row 5 row cell negative 3 end cell row 6 end table close parentheses plus 3 open parentheses table row cell negative 5 end cell row 3 row cell negative 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row 5 row cell negative 3 end cell row 6 end table close parentheses plus open parentheses table row cell negative 15 end cell row 9 row cell negative 18 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 10 end cell row 6 row cell negative 12 end cell end table close parentheses end cell end table

Diperoleh table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell a with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell b with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 10 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell i with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 6 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell j with rightwards arrow on top end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 12 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell k with rightwards arrow on top end cell end table

Oleh karena itu, jawaban yang tepat adalah B.

0

Roboguru

Diketahui titik  dan . Titik terletak pada garis  sehingga . Koordinat titik P adalah...

Pembahasan Soal:

Menentukan titik segaris (Kolinear)

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator stack A P with rightwards harpoon with barb upwards on top over denominator stack P B with rightwards harpoon with barb upwards on top end fraction end cell equals cell 3 over 1 end cell row cell stack A P with rightwards harpoon with barb upwards on top end cell equals cell 3 stack P B with rightwards harpoon with barb upwards on top end cell row cell P minus A end cell equals cell 3 open parentheses B minus P close parentheses end cell row cell P minus A end cell equals cell 3 B minus 3 P end cell row cell P plus 3 P end cell equals cell A plus 3 B end cell row cell 4 P end cell equals cell A plus 3 B end cell row cell 4 P end cell equals cell open parentheses table row 2 row 0 row 1 end table close parentheses plus 3 open parentheses table row 10 row 4 row 5 end table close parentheses end cell row blank equals cell open parentheses table row 2 row 0 row 1 end table close parentheses plus open parentheses table row 30 row 12 row 15 end table close parentheses end cell row blank equals cell open parentheses table row 32 row 12 row 16 end table close parentheses end cell row P equals cell 1 fourth open parentheses table row 32 row 12 row 16 end table close parentheses end cell row blank equals cell open parentheses table row 8 row 3 row 4 end table close parentheses end cell row blank equals cell open parentheses 8 comma 3 comma 4 close parentheses end cell end table 

Jadi, koordinat titik P equals open parentheses 8 comma 3 comma 4 close parentheses.

0

Roboguru

Diketahui titik  titik pangkal, dan titik-titik  dengan vektor posisi  dan . b.  Tentukan nilai  agar  segaris

Pembahasan Soal:

Misal dibentuk vektor sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack A B with rightwards arrow on top end cell equals cell stack O B with rightwards arrow on top minus stack O A with rightwards arrow on top end cell row blank equals cell open parentheses table row 4 row 2 end table close parentheses minus open parentheses table row 9 row cell negative 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 5 end cell row 12 end table close parentheses end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell stack B C with rightwards arrow on top end cell equals cell stack O C with rightwards arrow on top minus stack O B with rightwards arrow on top end cell row blank equals cell open parentheses table row m row cell negative 2 end cell end table close parentheses minus open parentheses table row 4 row 2 end table close parentheses end cell row blank equals cell open parentheses table row cell m minus 4 end cell row cell negative 4 end cell end table close parentheses end cell end table end style 

Dapat dilihat bahwa begin mathsize 14px style stack A B with rightwards arrow on top equals negative 3 stack B C with rightwards arrow on top end style, sehingga didapatkan:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 5 end cell equals cell negative 3 open parentheses m minus 4 close parentheses end cell row cell negative 5 end cell equals cell negative 3 m plus 12 end cell row cell negative 5 minus 12 end cell equals cell negative 3 m end cell row cell negative 17 end cell equals cell negative 3 m end cell row m equals cell fraction numerator negative 17 over denominator negative 3 end fraction end cell row m equals cell 17 over 3 end cell end table end style

Dengan demikian, nilai begin mathsize 14px style m end style agar begin mathsize 14px style A comma space B comma space dan space C end style segaris adalah begin mathsize 14px style 17 over 3 end style.

0

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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