Sudut antara vektor u=⎝⎛x1−3⎠⎞ dan vektor v⇀=⎝⎛−13−2⎠⎞ adalah 60∘. Nilai x= ...

Pertanyaan

Sudut antara vektor $\overrightarrow u=\begin{pmatrix}x\\1\\-3\end{pmatrix}$ dan vektor $\overset\rightharpoonup v=\begin{pmatrix}-1\\3\\-2\end{pmatrix}$ adalah $60^\circ$ . Nilai $x=$ ...

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Gunakan rumus sudut antara dua vektor.

Jika terdapat vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top , maka rumus sudut antara vektor keduanya yaitu:

$cos space alpha equals fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar times open vertical bar b with rightwards arrow on top close vertical bar end fraction$

Pada vektor u with rightwards arrow on top equals open parentheses table row x row 1 row cell negative 3 end cell end table close parentheses dan vektor v with rightwards harpoon with barb upwards on top equals open parentheses table row cell negative 1 end cell row 3 row cell negative 2 end cell end table close parentheses dengan sudut antara vektor 60 degree , diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator u with rightwards arrow on top times v with rightwards arrow on top over denominator open vertical bar u with rightwards arrow on top close vertical bar times open vertical bar v with rightwards arrow on top close vertical bar end fraction end cell row cell cos space 60 degree end cell equals cell fraction numerator x open parentheses negative 1 close parentheses plus 1 open parentheses 3 close parentheses plus open parentheses negative 3 close parentheses open parentheses negative 2 close parentheses over denominator square root of x squared plus 1 squared plus open parentheses negative 3 close parentheses squared end root times square root of open parentheses negative 1 close parentheses squared plus 3 squared plus open parentheses negative 2 close parentheses squared end root end fraction end cell row cell 1 half end cell equals cell fraction numerator negative x plus 3 plus 6 over denominator square root of x squared plus 1 plus 9 end root times square root of 1 plus 9 plus 4 end root end fraction end cell row cell 1 half end cell equals cell fraction numerator negative x plus 9 over denominator square root of x squared plus 10 end root times square root of 14 end fraction end cell row cell 2 open parentheses negative x plus 9 close parentheses end cell equals cell 1 open parentheses square root of x squared plus 10 end root times square root of 14 close parentheses end cell row cell negative 2 x plus 18 end cell equals cell square root of 14 x squared plus 140 end root end cell row cell open parentheses negative 2 x plus 18 close parentheses squared end cell equals cell open parentheses square root of 14 x squared plus 140 end root close parentheses squared end cell row cell 4 x squared minus 72 x plus 324 end cell equals cell 14 x squared plus 140 end cell row 0 equals cell 14 x squared minus 4 x squared plus 72 x minus 324 plus 140 end cell row 0 equals cell 10 x squared plus 72 x minus 184 end cell end table$

Kemudian faktorkan fungsi yang diperoleh untuk mengetahui nilai .

Diperoleh nilai yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 2 end cell equals 0 row x equals 2 end table

atau

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 x plus 46 end cell equals 0 row cell 5 x end cell equals cell negative 46 end cell row x equals cell negative 46 over 5 end cell end table

Karena pada pilihan jawaban hanya terdapat satu nilai, maka nilai x equals 2 .

Jadi, jawaban yang tepat adalah E.