A small particle moves to position $$5\hat{i} - 2\hat{j} + \hat{k}$$ from its initial position $$2\hat{i} + 3\hat{j} - 4\hat{k}$$ under the action of force $$5\hat{i} + 2\hat{j} + 7\hat{k}$$ N. The value of work done will be _____ J.

We have initial position $$\vec{r_i} = 2\hat{i} + 3\hat{j} - 4\hat{k}$$, final position $$\vec{r_f} = 5\hat{i} - 2\hat{j} + \hat{k}$$, and force $$\vec{F} = 5\hat{i} + 2\hat{j} + 7\hat{k}$$ N.

The displacement vector is:

$$\vec{d} = \vec{r_f} - \vec{r_i} = (5-2)\hat{i} + (-2-3)\hat{j} + (1-(-4))\hat{k} = 3\hat{i} - 5\hat{j} + 5\hat{k}$$

Now, work done is the dot product of force and displacement:

$$W = \vec{F} \cdot \vec{d} = (5)(3) + (2)(-5) + (7)(5)$$

$$W = 15 - 10 + 35 = 40 \text{ J}$$

So, the answer is $$40$$ J.