The torque of a force $$5\hat{i} + 3\hat{j} - 7\hat{k}$$ about the origin is $$\vec{\tau}$$. If the force acts on a particle whose position vector is $$2\hat{i} + 2\hat{j} + \hat{k}$$, then the value of $$\vec{\tau}$$ will be

We have the position vector $$\vec{r} = 2\hat{i} + 2\hat{j} + \hat{k}$$ and the force $$\vec{F} = 5\hat{i} + 3\hat{j} - 7\hat{k}$$. The torque about the origin is $$\vec{\tau} = \vec{r} \times \vec{F}$$.

We compute the cross product using the determinant formula: $$\vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 2 & 1 \\ 5 & 3 & -7 \end{vmatrix}$$

The $$\hat{i}$$ component is $$(2)(-7) - (1)(3) = -14 - 3 = -17$$. The $$\hat{j}$$ component is $$-\left[(2)(-7) - (1)(5)\right] = -(-14 - 5) = 19$$. The $$\hat{k}$$ component is $$(2)(3) - (2)(5) = 6 - 10 = -4$$.

Therefore $$\vec{\tau} = -17\hat{i} + 19\hat{j} - 4\hat{k}$$.

Hence, the correct answer is Option 3.