Two particles A and B, move with constant velocities `vec(v_(1))" and "vec(v_(2))`. At the initial moment their position vectors are `vec(r_(1))" and "vec(r_(2))` respectively. The condition for particle A and B for their collision is

Options

A. `vec(r_(1))-vec(r_(2))=vec(v_(1))-vec(v_(2))`
B. `(vec(r_(1))-vec(r_(2)))/(|vec(r_(1))-vec(r_(2))|)=(vec(v_(2))-vec(v_(1)))/(|vec(v_(2))-vec(v_(1))|)`
C. `vec(r_(1)).vec(v_(1))=vec(r_(2)).vec(v_(2))`
D. `vec(r_(1))xx vec(v_(1))=vec(r_(2))xx vec(v_(2))`

Answer

To determine the condition for the collision of two particles A and B moving with constant velocities, we need to analyze their motion using the equations of motion in vector form.

### Step-by-Step Solution:

1. **Define the Position Vectors**:
Let the position vectors of particles A and B at time \( t \) be given by:
\[ \vec{r}_A(t) = \vec{r}_1 + \vec{v}_1 t \]
\[ \vec{r}_B(t) = \vec{r}_2 + \vec{v}_2 t \]

2. **Set the Condition for Collision**:
For the two particles to collide, their position vectors must be equal at the same time \( t \):
\[ \vec{r}_A(t) = \vec{r}_B(t) \]

3. **Equate the Position Vectors**:
Substituting the expressions for \( \vec{r}_A(t) \) and \( \vec{r}_B(t) \):
\[ \vec{r}_1 + \vec{v}_1 t = \vec{r}_2 + \vec{v}_2 t \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ \vec{r}_1 - \vec{r}_2 = (\vec{v}_2 - \vec{v}_1) t \]

5. **Solving for Time \( t \)**:
If we denote \( \vec{r}_{AB} = \vec{r}_1 - \vec{r}_2 \) and \( \vec{v}_{AB} = \vec{v}_2 - \vec{v}_1 \), we can express the time \( t \) when the collision occurs as:
\[ t = \frac{\vec{r}_{AB}}{\vec{v}_{AB}} \]
This equation implies that for the particles to collide, the vector \( \vec{r}_{AB} \) must be parallel to \( \vec{v}_{AB} \).

6. **Condition for Collision**:
Therefore, the condition for the collision of particles A and B is that the vector \( \vec{r}_{AB} \) must be parallel to the vector \( \vec{v}_{AB} \):
\[ \vec{r}_{AB} \times \vec{v}_{AB} = 0 \]

### Final Condition:
The final condition for the collision of the two particles A and B is:
\[ (\vec{r}_1 - \vec{r}_2) \times (\vec{v}_2 - \vec{v}_1) = 0 \]