Pertanyaan

Diketahui vektor u = ⎝ ⎛ 2 − 1 1 ⎠ ⎞ dan v → = ⎝ ⎛ 7 0 1 ⎠ ⎞ . Tentukan sudut antara u dan v .

Diketahui vektor $u = ⎝ ⎛ 2 - 1 1 ⎠ ⎞$ dan $v \to = ⎝ ⎛ 701 ⎠ ⎞$ . Tentukan sudut antara $u$ dan $v$ .

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Jawaban terverifikasi

Jawaban

besarsudut antara dan adalah .

besar sudut antara begin mathsize 14px style u with rightwards arrow on top end style

dan

begin mathsize 14px style v with rightwards arrow on top end style

adalah

begin mathsize 14px style 30 degree end style

Pembahasan

Jika vektor dan , makahasil kali titik ( dot product )dari dua vektor adalah dan dengan adalah sudut di antara kedua vektor. Perhatikan bahwa dan Akibatnya Jadi, besarsudut antara dan adalah .

Jika vektor begin mathsize 14px style u with rightwards arrow on top equals open parentheses table row cell x subscript 1 end cell row cell y subscript 1 end cell row cell z subscript 1 end cell end table close parentheses end style dan begin mathsize 14px style v with rightwards arrow on top equals open parentheses table row cell x subscript 2 end cell row cell y subscript 2 end cell row cell z subscript 2 end cell end table close parentheses end style , maka hasil kali titik (dot product) dari dua vektor adalah

begin mathsize 14px style u with rightwards arrow on top times v with rightwards arrow on top equals x subscript 1 x subscript 2 plus y subscript 1 y subscript 2 plus z subscript 1 z subscript 2 end style

dan

begin mathsize 14px style u with rightwards arrow on top times v with rightwards arrow on top equals open vertical bar u with rightwards arrow on top close vertical bar open vertical bar v with rightwards arrow on top close vertical bar cos theta end style

dengan begin mathsize 14px style theta end style adalah sudut di antara kedua vektor.

Perhatikan bahwa

dan

Akibatnya

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell u with rightwards arrow on top times v with rightwards arrow on top end cell equals cell vertical line u with rightwards arrow on top vertical line vertical line v with rightwards arrow on top vertical line space cos space theta end cell row cell open parentheses table row 2 row cell negative 1 end cell row 1 end table close parentheses times open parentheses table row 7 row 0 row 1 end table close parentheses end cell equals cell open parentheses square root of 6 close parentheses open parentheses 5 square root of 2 close parentheses space cos space theta end cell row cell 14 plus 0 plus 1 end cell equals cell 5 square root of 12 space cos space theta end cell row 15 equals cell 10 square root of 3 space cos space theta end cell row cell fraction numerator 15 over denominator 10 square root of 3 end fraction end cell equals cell cos space theta end cell row cell fraction numerator 15 over denominator 10 square root of 3 end fraction times fraction numerator square root of 3 over denominator square root of 3 end fraction end cell equals cell cos space theta end cell row cell 15 over 30 square root of 3 end cell equals cell cos space theta end cell row cell 1 half square root of 3 end cell equals cell cos space theta end cell row cell 30 degree end cell equals theta end table end style$

Jadi, besar sudut antara begin mathsize 14px style u with rightwards arrow on top end style dan begin mathsize 14px style v with rightwards arrow on top end style adalah begin mathsize 14px style 30 degree end style .