If `vec F` is the force acting in a particle having position vector `vec r` and `vec tau` be the torque of this force about the origin, then

Options

A. `vec r. vec tau ne 0 and vec F. vec tau = 0`
B. `vec r. vec tau gt 0 and vec F. vec tau lt 0`
C. `vec r. vec tau = 0 and vec F. vec tau = 0`
D. `vec r. vec tau = 0 and vec F. vec tau ne 0`

Explanation

To solve the problem, we need to understand the relationship between force, position vector, and torque in the context of rotational dynamics.

### Step-by-Step Solution:

1. **Understanding Torque**:
Torque (\( \vec{\tau} \)) is defined as the cross product of the position vector (\( \vec{r} \)) and the force vector (\( \vec{F} \)). Mathematically, it can be expressed as:
\[ \vec{\tau} = \vec{r} \times \vec{F} \]

2. **Direction of Torque**:
The direction of the torque vector is given by the right-hand rule, which states that if you curl the fingers of your right hand from the position vector (\( \vec{r} \)) towards the force vector (\( \vec{F} \)), your thumb will point in the direction of the torque vector (\( \vec{\tau} \)).

3. **Perpendicular Nature**:
Since torque is a cross product, it is always perpendicular to both the position vector (\( \vec{r} \)) and the force vector (\( \vec{F} \)). This means:
\[ \vec{\tau} \perp \vec{r} \quad \text{and} \quad \vec{\tau} \perp \vec{F} \]

4. **Physical Interpretation**:
The torque represents the tendency of a force to cause rotation about an axis. The larger the torque, the greater the potential for rotational motion.

5. **Units of Torque**:
The SI unit of torque is Newton-meter (N·m), which comes from the units of force (N) and distance (m).

### Final Statement:
Thus, we conclude that the torque \( \vec{\tau} \) due to a force \( \vec{F} \) acting on a particle at position \( \vec{r} \) about the origin is given by:
\[ \vec{\tau} = \vec{r} \times \vec{F} \]
and it is always perpendicular to both \( \vec{r} \) and \( \vec{F} \).