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SIMAK UI 2010

Solusi SIMAK UI 2010 Kode 509 [IPA] – Nomor 6


Vektor\ \vec{a},\ \vec{b},\ \vec{c},\ adalah\ vektor-vektor\ unit\ yang\ masing-masing\\membentuk\ sudut\ 60^{\circ}\ dengan\ vektor\ lainnya,\ maka\ \big(\vec{a}-\vec{b}\big)\bullet \big(\vec{b}-\vec{c}\big)\ adalah\ ....
(A)\quad -\frac{1}{4}
(B)\quad -\frac{1}{2}
(C)\quad \frac{-\sqrt{3}}{2}
(D)\quad \frac{1}{4}
(E)\quad \frac{1}{2}
Jawaban: B

\begin{array} {lcl} \big(\vec{a}-\vec{b}\big)\bullet \big(\vec{b}-\vec{c}\big)&=&\vec{a}\bullet \vec{b}-\vec{a}\bullet \vec{c}-\vec{b}\bullet \vec{b}+\vec{b}\bullet \vec{c}\\&=&\vec{a}\bullet \vec{b}-\vec{a}\bullet \vec{c}-\left|\vec{b}\right|^2+\vec{b}\bullet \vec{c}\quad .................\quad (*)\end{array}
Dari unsur-unsur yang diketahui pada soal diperoleh:

\vec{a}\bullet \vec{b}=\left|\vec{a}\right|\times \left|\vec{b}\right|\times cos60^{\circ}=1\times 1\times \frac{1}{2}=\frac{1}{2}
\vec{a}\bullet \vec{c}=\left|\vec{a}\right|\times \left|\vec{c}\right|\times cos60^{\circ}=1\times 1\times \frac{1}{2}=\frac{1}{2}
\vec{b}\bullet \vec{c}=\left|\vec{b}\right|\times \left|\vec{c}\right|\times cos60^{\circ}=1\times 1\times \frac{1}{2}=\frac{1}{2}
sehingga (*) menjadi:

\big(\vec{a}-\vec{b}\big)\bullet \big(\vec{b}-\vec{c}\big)=\frac{1}{2}-\frac{1}{2}-1^2+\frac{1}{2}=-\frac{1}{2}

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