In a 3 dimensional space, two particles are moving with uniform speeds of `6 m//s` and `8 m//s` along two arbitrary curves. The speed of one particle, as observed by other can be `:`

Options

A. zero
B. `15m//s`
C. `13m//s`
D. depends upon the actual shape of curves and the locations of two particles at some instant.

Answer

To solve the problem of finding the speed of one particle as observed by the other in a 3-dimensional space, we can use the concept of relative velocity.

### Step-by-Step Solution:

1. **Identify the Speeds of the Particles**:
Let the speed of particle A be \( v_A = 6 \, \text{m/s} \) and the speed of particle B be \( v_B = 8 \, \text{m/s} \).

2. **Understand Relative Velocity**:
The speed of one particle as observed by the other can be calculated using the formula for relative velocity. If two objects are moving with velocities \( \vec{v}_A \) and \( \vec{v}_B \), the relative velocity of A with respect to B is given by:
\[ \vec{v}_{AB} = \vec{v}_A - \vec{v}_B \]

3. **Magnitude of Relative Velocity**:
The magnitude of the relative velocity can be calculated using the formula:
\[ v_{AB} = |\vec{v}_{AB}| = |\vec{v}_A - \vec{v}_B| \]
Since the particles are moving along arbitrary curves, we need to consider the angle \( \theta \) between their paths.

4. **Using the Law of Cosines**:
The magnitude of the relative velocity can also be expressed using the law of cosines:
\[ v_{AB} = \sqrt{v_A^2 + v_B^2 - 2 v_A v_B \cos \theta} \]
where \( \theta \) is the angle between the directions of the two velocities.

5. **Calculate the Range of Possible Speeds**:
- If the particles are moving in the same direction (\( \theta = 0 \)):
\[ v_{AB} = |v_A - v_B| = |6 - 8| = 2 \, \text{m/s} \]
- If the particles are moving in opposite directions (\( \theta = 180^\circ \)):
\[ v_{AB} = v_A + v_B = 6 + 8 = 14 \, \text{m/s} \]
- For any angle \( \theta \) between \( 0 \) and \( 180^\circ \), the speed of one particle as observed by the other can vary between \( 2 \, \text{m/s} \) and \( 14 \, \text{m/s} \).

### Conclusion:
The speed of one particle as observed by the other can vary from a minimum of \( 2 \, \text{m/s} \) to a maximum of \( 14 \, \text{m/s} \).