Animated Solution for Physics - Rotational Motion: A force of $-F\hat{k}$ acts on $O$, the origin of the coordinate system. The torque about the point $(1,-1)$ is

Welcome to this interesting problem on torque! The question presents a slight language twist. While the text literally states that the force acts at the origin and asks for the torque about the point $(1,-1)$, the official solution interprets it differently. Based on the provided answer key, we are actually calculating the torque of a force acting at the point $(1,-1)$ about the origin.

Let's visualize the setup based on this intended meaning. Our position vector $\mathbf{r}$ points from the origin to the point of application, giving us $\mathbf{r}=\hat{i}-\hat{j}$. The force vector is given as $\mathbf{F}=-F\hat{k}$.

To find the torque, we use the fundamental definition involving the cross product of the position vector and the force vector:

Substituting our known vectors into this equation, we get:

Now, we distribute the cross product. It is crucial to maintain the strict order of the vectors, as the cross product is anti-commutative.

Recall your cyclic rules for unit vectors. Moving counter-clockwise on the $\hat{i},\hat{j},\hat{k}$ circle yields negative results, while moving clockwise yields positive results. Therefore:

Plugging these back into our torque expression:

Simplifying the expression, the minus signs cancel out beautifully, leaving us with our final answer:

A quick note for the astute student: if the force actually acted at the origin and we wanted the torque about the point $(1,-1)$ as literally stated, the position vector of the force relative to the pivot would be reversed ($-\hat{i}+\hat{j}$). This would result in a torque of $-F(\hat{i}+\hat{j})$, which corresponds to option (b). Always be prepared to interpret the intended meaning in competitive exams when discrepancies arise!