Diketahui koordinat \( P(-3,2,1) \) dan \( Q(7,-3,11) \). Jika titik R membagi PQ dengan perbandingan \( \overrightarrow{PR}: \overrightarrow{RQ} = 3:2 \) maka \( \overrightarrow{PR} \cdot \overrightarrow{RQ} = \cdots \)
- \( 54 \)
- \( 36 \)
- \( 30 \)
- \( 24 \)
- \( 20 \)
Pembahasan:
Karena \( \overrightarrow{PR}: \overrightarrow{RQ} = 3:2 \) maka kita peroleh berikut:
\begin{aligned} 2 \overrightarrow{PR} &= 3 \overrightarrow{RQ} \\[8pt] 2 (\vec{r}-\vec{p}) &= 3 (\vec{q}-\vec{r}) \\[8pt] 2\vec{r}-2\vec{p} &= 3\vec{q}-3\vec{r} \\[8pt] 5\vec{r} &= 2\vec{p}+3\vec{q} \\[8pt] 5 \vec{r} &= 2(7, -3, 11)+3(-3,2,1) \\[8pt] 5 \vec{r} &= (14, -6, 22)+(-9, 6, 3) \\[8pt] 5 \vec{r} &= (5, 0, 25) \Rightarrow \vec{r} = (1,0,5) \\[8pt] \overrightarrow{PR} &= \vec{r}-\vec{p} = (1,0,5) - (-3,2,1) \\[8pt] &= (4, -2, 4) \\[8pt] \overrightarrow{RQ} &= \vec{q}-\vec{r} = (7, -3, 11)-(1,0,5) \\[8pt] &= (6,-3,6) \end{aligned}
Selanjutnya, dari hasil di atas, kita peroleh:
\begin{aligned} \overrightarrow{PR} \cdot \overrightarrow{RQ} &= (4, -2, 4) \cdot (6, -3, 6) \\[8pt] &= (4)(6)+(-2)(-3)+(4)(6) \\[8pt] &= 24+6+24 \\[8pt] &= 54 \end{aligned}
Jawaban A.