Two particles, 1 and 2, move with constant velocities `v_1` and `v_2`. At the initial moment their radius vectors are equal to `r_1` and `r_2`. How must these four vectors be interrelated for the particles to collide?

Explanation

To determine how the radius vectors and velocities of two particles must be interrelated for them to collide, we can follow these steps:

### Step 1: Define the positions of the particles
Let the position of particle 1 at time \( t \) be given by:
\[ \mathbf{r}_1(t) = \mathbf{r}_1 + \mathbf{v}_1 t \]
And for particle 2:
\[ \mathbf{r}_2(t) = \mathbf{r}_2 + \mathbf{v}_2 t \]

### Step 2: Set the positions equal for collision
For the two particles to collide, their positions must be equal at some time \( t \):
\[ \mathbf{r}_1 + \mathbf{v}_1 t = \mathbf{r}_2 + \mathbf{v}_2 t \]

### Step 3: Rearrange the equation
Rearranging the equation gives:
\[ \mathbf{r}_1 - \mathbf{r}_2 = (\mathbf{v}_2 - \mathbf{v}_1) t \]

### Step 4: Express time \( t \)
From the rearranged equation, we can express time \( t \) as:
\[ t = \frac{\mathbf{r}_1 - \mathbf{r}_2}{\mathbf{v}_2 - \mathbf{v}_1} \]
This equation indicates that for the particles to collide, the vector difference in their initial positions must be proportional to the vector difference in their velocities.

### Step 5: Conditions for collision
For the above equation to yield a valid time \( t \), the following conditions must be satisfied:
1. The vector \( \mathbf{v}_2 - \mathbf{v}_1 \) should not be zero (i.e., the velocities must be different).
2. The direction of \( \mathbf{r}_1 - \mathbf{r}_2 \) must be the same as the direction of \( \mathbf{v}_2 - \mathbf{v}_1 \).

### Conclusion
Thus, the relationship between the vectors for the particles to collide can be summarized as:
- The difference in their initial position vectors \( \mathbf{r}_1 - \mathbf{r}_2 \) must be parallel to the difference in their velocity vectors \( \mathbf{v}_2 - \mathbf{v}_1 \).