Diketahui \( A(1,2,3), \ B(3,3,1) \) dan \(C(7,5,-3)\). Jika A, B, dan C segaris (kolinear), maka perbandingan \( \overrightarrow{AB}:\overrightarrow{BC} \) adalah… (SBMPTN 2014)
- \(1:2\)
- \(2:1\)
- \(2:5\)
- \(5:7\)
- \(7:5\)
Pembahasan:
Perbandingan \( \overrightarrow{AB}:\overrightarrow{BC} \) dapat dicari sebagai berikut:
\begin{aligned} \overrightarrow{AB} &= B-A = (3,3,1)-(1,2,3) \\[8pt] &= (2,1,-2) \\[8pt] |\overrightarrow{AB}| &= \sqrt{2^2+1^2+(-2)^2} = \sqrt{4+1+4} \\[8pt] &= \sqrt{9} = 3 \\[8pt] \overrightarrow{BC} &= C-B = (7,5,-3)-(3,3,1) \\[8pt] &= (4,2,-4) \\[8pt] |\overrightarrow{AB}| &= \sqrt{4^2+2^2+(-4)^2} = \sqrt{16+4+16} \\[8pt] &= \sqrt{36} = 6 \\[8pt] \overrightarrow{AB}:\overrightarrow{BC} &= |\overrightarrow{AB}|:|\overrightarrow{BC}| = 3:6 \\[8pt] &= 1:2 \end{aligned}
Jawaban A.