Jika vektor \( \vec{a} = 10 \hat{i} + \hat{j} + \hat{k} \) dan \( \vec{b} = 3 \hat{i} + 2 \hat{j} + 2 \hat{k} \). Proyeksi ortogonal untuk vektor \( \vec{a} \) pada \( \vec{b} \) adalah…
- \( 4 \hat{i} + 2\hat{j} + 3\hat{k} \)
- \( -4 \hat{i} - \hat{j} + 3\hat{k} \)
- \( 6 \hat{i} + 3 \hat{j} + 2 \hat{k} \)
- \( \hat{i} + 4 \hat{j} - 6 \hat{k} \)
- \( 6 \hat{i} + 4 \hat{j} + 4 \hat{k} \)
Pembahasan:
Misalkan \( \vec{c} \) adalah proyeksi ortogonal vektor \( \vec{a} \) pada \( \vec{b} \), maka \( \vec{c} \), yaitu:
\begin{aligned} \vec{c} &= \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \cdot \vec{b} = \left( \frac{(10,1,1) \cdot (3,2,2)}{\left( \sqrt{3^2+2^2+2^2} \right)^2} \right) \cdot (3 \hat{i} + 2 \hat{j} + 2 \hat{k}) \\[8pt] &= \left( \frac{(10)(3)+(1)(2)+(1)(2)}{9+4+4} \right) \cdot (3 \hat{i} + 2 \hat{j} + 2 \hat{k}) \\[8pt] &= \left( \frac{30+2+2}{17} \right) \cdot (3 \hat{i} + 2 \hat{j} + 2 \hat{k}) \\[8pt] &= \frac{34}{17} \cdot (3 \hat{i} + 2 \hat{j} + 2 \hat{k}) \\[8pt] &= 2 \cdot (3 \hat{i} + 2 \hat{j} + 2 \hat{k}) \\[8pt] &= 6 \hat{i} + 4 \hat{j} + 4 \hat{k} \end{aligned}
Jawaban E.