If is the force acting on a particle having position vector and be the torque of this force about the origin, then-

Options

A. `vecr . vectau` and `vecF. vectau ne 0`
B. `vecr . vectau` and `vecF. vectau = 0`
C. `vecr.vectau gt0` and `vecF. vectau lt 0`
D. `vecr.vectau = 0` and ` vecF. vectau =0`

Answer

To solve the problem, we need to understand the relationship between the force acting on a particle, its position vector, and the torque produced by that force about the origin.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( \mathbf{r} \) be the position vector of the particle.
- Let \( \mathbf{F} \) be the force acting on the particle.
- The torque \( \mathbf{\tau} \) about the origin due to the force \( \mathbf{F} \) is given by the cross product:
\[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \]

2. **Understanding the Dot Product:**
- The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined as:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \]
- Where \( \theta \) is the angle between the two vectors.

3. **Evaluate \( \mathbf{r} \cdot \mathbf{\tau} \):**
- We can express \( \mathbf{r} \cdot \mathbf{\tau} \) as:
\[ \mathbf{r} \cdot \mathbf{\tau} = \mathbf{r} \cdot (\mathbf{r} \times \mathbf{F}) \]
- The dot product of a vector with a cross product of itself and another vector is always zero:
\[ \mathbf{r} \cdot (\mathbf{r} \times \mathbf{F}) = 0 \]
- Therefore, \( \mathbf{r} \cdot \mathbf{\tau} = 0 \).

4. **Evaluate \( \mathbf{F} \cdot \mathbf{\tau} \):**
- Now, consider \( \mathbf{F} \cdot \mathbf{\tau} \):
\[ \mathbf{F} \cdot \mathbf{\tau} = \mathbf{F} \cdot (\mathbf{r} \times \mathbf{F}) \]
- Again, using the property of the dot product with a cross product:
\[ \mathbf{F} \cdot (\mathbf{r} \times \mathbf{F}) = 0 \]
- Therefore, \( \mathbf{F} \cdot \mathbf{\tau} = 0 \).

5. **Conclusion:**
- From the above evaluations, we conclude that:
\[ \mathbf{r} \cdot \mathbf{\tau} = 0 \quad \text{and} \quad \mathbf{F} \cdot \mathbf{\tau} = 0 \]
- This means that both dot products are equal to zero.

### Final Answer:
- \( \mathbf{r} \cdot \mathbf{\tau} = 0 \) and \( \mathbf{F} \cdot \mathbf{\tau} = 0 \).