Diketahui vektor \( \vec{a} = 2\hat{i}-2p\hat{j}+4\hat{k} \) dan \( \vec{b} = \hat{i}-3\hat{j}+4\hat{k} \). Jika panjang proyeksi vektor \( \vec{a} \) pada \( \vec{b} \) adalah \( \frac{6}{ \sqrt{26} } \), maka nilai \(p = \cdots \) (UN 2014)
- -3
- -2
- -1
- 1
- 3
Pembahasan:
Nilai \(p\) dapat dicari sebagai berikut:
\begin{aligned} \frac{6}{\sqrt{26}} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \Leftrightarrow \frac{6}{\sqrt{26}} &= \frac{(2,-2p,4) \cdot (1,-3,4) }{\sqrt{1+(-3)^2+4^2}} \\[8pt] \frac{6}{\sqrt{26}} &= \frac{(2)(1)+(-2p)(-3)+(4)(4)}{\sqrt{1+9+16}} \\[8pt] \frac{6}{\sqrt{26}} &= \frac{2+6p+16}{\sqrt{26}} \\[8pt] 6 &= 2+6p+16 \\[8pt] 6 &= 6p+18 \\[8pt] 6p &= 6-18 \\[8pt] p &= \frac{-12}{6} = -2 \end{aligned}
Jawaban B.