Explanation: \(\text{Solve: এখানে, সরণ } \vec{r} = (6 - 2)\hat{i} + (8 - \alpha)\hat{j} + (3\alpha - 4)\hat{k} \\ \implies কাজ \, W = \vec{F} \cdot \vec{r} \\ = (\hat{i} + 2\hat{j} + 4\hat{k}) \cdot [4\hat{i} + (8 - \alpha)\hat{j} + (3\alpha - 4)\hat{k}] \\ = 4a + 2(8-\alpha) + 4(3\alpha-4) \\ = 4a + 16 - 2\alpha + 12\alpha - 16 \\ = 14a \\ \text{যেহেতু } 14a = 42 \implies a = 3 \\ \text{Ans. (C)}\)
Another Explanation (5):
এখানে, কৃতকাজ \( W = 42 \) J এবং বল \( \vec{F} = \alpha \hat{i} + 2\hat{j} + 4\hat{k} \) N। বস্তুটির সরণ \( \vec{d} \) নির্ণয় করতে হবে।
সরণ, \( \vec{d} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k} \)
এখানে, \( (x_1, y_1, z_1) = (2, \alpha, 4) \) এবং \( (x_2, y_2, z_2) = (6, 8, 3\alpha) \)
সুতরাং, \( \vec{d} = (6 - 2)\hat{i} + (8 - \alpha)\hat{j} + (3\alpha - 4)\hat{k} \)
\( \vec{d} = 4\hat{i} + (8 - \alpha)\hat{j} + (3\alpha - 4)\hat{k} \)
আমরা জানি, কাজ \( W = \vec{F} \cdot \vec{d} \)
সুতরাং, \( 42 = (\alpha \hat{i} + 2\hat{j} + 4\hat{k}) \cdot (4\hat{i} + (8 - \alpha)\hat{j} + (3\alpha - 4)\hat{k}) \)
ডট গুণফল করলে পাই,
\( 42 = 4\alpha + 2(8 - \alpha) + 4(3\alpha - 4) \)
\( 42 = 4\alpha + 16 - 2\alpha + 12\alpha - 16 \)
\( 42 = 14\alpha \)
\( \alpha = \frac{42}{14} \)
\( \alpha = 3 \)
অতএব, \( \alpha \) এর মান 3। 🎉
