Diketahui tiga vektor yaitu \( \vec{a}=3x \hat{i} + x \hat{j}-4 \hat{k}, \ \vec{b}=-2 \hat{i} + 4 \hat{j} + 5 \hat{k} \) dan \( \vec{c}=-3 \hat{i} + 2 \hat{j} + \hat{k} \). Jika \(\vec{a}\) tegak lurus pada \(\vec{b}\) maka \( \vec{a}-\vec{c} \) adalah… (UMPTN 1993)
- \( -33 \hat{i} - 8 \hat{j} - 5 \hat{k} \)
- \( -27 \hat{i} - 8 \hat{j} - 5 \hat{k} \)
- \( -27 \hat{i} - 12 \hat{j} - 5 \hat{k} \)
- \( 33 \hat{i} - 12 \hat{j} - 5 \hat{k} \)
- \( 27 \hat{i} - 12 \hat{j} - 5 \hat{k} \)
Pembahasan:
Karena \( \vec{a} \) tegak lurus pada \( \vec{b} \) maka kita peroleh:
\begin{aligned} \vec{a} \cdot \vec{b} &= 0 \\[8pt] (3x \hat{i} + x \hat{j}-4 \hat{k}) \cdot (-2 \hat{i} + 4 \hat{j} + 5 \hat{k}) &= 0 \\[8pt] (3x)(-2)+(x)(4)+(-4)(5) &= 0 \\[8pt] -6x+4x-20 &= 0 \\[8pt] -2x &= 20 \\[8pt] x &= -10 \end{aligned}
Untuk \(x=-10\), maka kita dapatkan hasil berikut:
\begin{aligned} \vec{a} &= 3x \hat{i} + x \hat{j}-4 \hat{k} = -30 \hat{i} - 10 \hat{j}-4 \hat{k} \\[8pt] \vec{c} &= -3 \hat{i} + 2 \hat{j} + \hat{k} \\[8pt] \vec{a}-\vec{c} &= (-30 \hat{i} - 10 \hat{j}-4 \hat{k})-(-3 \hat{i} + 2 \hat{j} + \hat{k}) \\[8pt] &= (-30+3) \hat{i} + (-10-2) \hat{j} + (-4-1) \hat{k} \\[8pt] &= -27 \hat{i} - 12 \hat{j} - 5 \hat{k} \end{aligned}
Jawaban C.