Vektor-vektor \( u, v \) dan \(w\) tak nol dan \( |u|=|v| \). Jika \( |v-w|=|u-w| \), maka...
- \( u \cdot v = |w| \)
- \( w = \frac{2u+3v}{5} \)
- \( |u-w|=|v| \)
- \( u-v \) tegak lurus \(w\)
- \(u+w \) tegak lurus \(w\)
(SBMPTN 2014)
Pembahasan:
Diketahui \( |v-w|=|u-w| \) sehingga kita peroleh berikut:
\begin{aligned} |v-w| &= |u-w| \\[8pt] \sqrt{(v-w)^2} &= \sqrt{(u-w)^2} \\[8pt] (v-w)^2 &= (u-w)^2 \\[8pt] v \cdot v - 2v \cdot w + w \cdot w &= u \cdot u - 2 u \cdot w + w \cdot w \\[8pt] |v|^2-2 v \cdot w + |w|^2 &= |u|-2 u \cdot w + |w|^2; \qquad \text{Ingat:} \ |u|=|v| \\[8pt] 2v \cdot w &= 2 u \cdot w \\[8pt] v \cdot w &= u \cdot w \\[8pt] u \cdot w-v \cdot w &= 0 \\[8pt] (u \cdot v) \cdot w &= 0 \end{aligned}
Karena \( (u \cdot v) \cdot w \) sama dengan nol, maka vektor \( (u-v) \) tegak lurus vektor \(w\).
Jawaban D.