Diketahui vektor a=⎝⎛x12⎠⎞ dan b=⎝⎛23−1⎠⎞ dan panjang proyeksi vektor a pada vektor b adalah 7114. Sudut antara dan adalah α, maka cos α= ...

Pertanyaan

Diketahui vektor $\overrightarrow{\text{a}}=\begin{pmatrix}x\\1\\2\end{pmatrix}$ dan $\overrightarrow{\text{b}}=\begin{pmatrix}2\\3\\-1\end{pmatrix}$ dan panjang proyeksi vektor $\overrightarrow{\text{a}}$ pada vektor $\overrightarrow{\text{b}}$ adalah $\frac17\sqrt{14}$ . Sudut antara stack text a end text with rightwards arrow on top dan stack text b end text with rightwards arrow on top adalah $\alpha$ , maka $\cos\;\alpha=\;...$

H. Eka

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

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Pembahasan

Panjang vektor a with rightwards arrow on top equals x i with rightwards arrow on top plus y j with rightwards arrow on top plus z k with rightwards arrow on top , yaitu open vertical bar a with rightwards arrow on top close vertical bar equals square root of x squared plus y squared plus z squared end root

Jika vektor a with rightwards arrow on top equals a subscript 1 i with rightwards arrow on top plus a subscript 2 j with rightwards arrow on top plus a subscript 3 k with rightwards arrow on top dan b with rightwards arrow on top equals b subscript 1 i with rightwards arrow on top plus b subscript 2 j with rightwards arrow on top plus b subscript 3 k with rightwards arrow on top , maka

a with rightwards arrow on top times b with rightwards arrow on top equals a subscript 1 times b subscript 1 plus a subscript 2 times b subscript 2 plus a subscript 3 times b subscript 3

Jika vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top membentuk sudut alpha , maka berlaku:

a with rightwards arrow on top times b with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space cos space alpha

Panjang proyeksi vektor a with rightwards arrow on top pada vektor b with rightwards arrow on top dapat ditentukan menggunakan rumus berikut.

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

Berdasarkan konsep di atas, nilai dapat ditentukan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar c with rightwards arrow on top close vertical bar end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator open parentheses table row x row 1 row 2 end table close parentheses open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses over denominator square root of 2 squared plus 3 squared plus open parentheses negative 1 close parentheses squared end root end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator 2 x plus 1 times 3 plus 2 open parentheses negative 1 close parentheses over denominator square root of 4 plus 9 plus 1 end root end fraction end cell row cell 1 over 7 square root of 14 end cell equals cell fraction numerator 2 x plus 1 over denominator square root of 14 end fraction end cell row cell 1 over 7 square root of 14 times square root of 14 end cell equals cell 2 x plus 1 end cell row cell 14 over 7 end cell equals cell 2 x plus 1 end cell row 2 equals cell 2 x plus 1 end cell row cell 2 x end cell equals 1 row x equals cell 1 half end cell end table$

Diperoleh x equals 1 half sehingga a with rightwards arrow on top equals open parentheses table row cell 1 half end cell row 1 row 2 end table close parentheses

Nilai cos space alpha adalah sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator open vertical bar a with rightwards arrow on top close vertical bar end fraction times fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of open parentheses begin display style 1 half end style close parentheses squared plus 1 squared plus 2 squared end root end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 1 over denominator square root of begin display style 21 over 4 end style end root end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 1 over denominator begin display style 1 half end style square root of 21 end fraction times 1 over 7 square root of 14 end cell row blank equals cell fraction numerator 2 over denominator square root of 21 end fraction times 1 over 7 square root of 14 end cell row blank equals cell 2 over 21 square root of 6 end cell end table$

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