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?

Options

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

Explanation

To determine how the vectors \( \mathbf{r_1} \), \( \mathbf{r_2} \), \( \mathbf{v_1} \), and \( \mathbf{v_2} \) must be interrelated for the two particles to collide, we can follow these steps:

### Step 1: Define the position of the particles at the time of collision
Let the collision occur at point \( A \) with position vector \( \mathbf{R_A} \). At time \( t \), the position vectors of the two particles can be expressed as:
\[ \mathbf{R_A} = \mathbf{r_1} + \mathbf{v_1} t \]
\[ \mathbf{R_A} = \mathbf{r_2} + \mathbf{v_2} t \]

### Step 2: Set the equations equal to each other
Since both expressions equal the position vector \( \mathbf{R_A} \), we can set them equal:
\[ \mathbf{r_1} + \mathbf{v_1} t = \mathbf{r_2} + \mathbf{v_2} t \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ \mathbf{r_1} - \mathbf{r_2} = (\mathbf{v_2} - \mathbf{v_1}) t \]

### Step 4: Solve for time \( t \)
From the rearranged equation, we can express \( t \) as:
\[ t = \frac{\mathbf{r_1} - \mathbf{r_2}}{\mathbf{v_2} - \mathbf{v_1}} \]
This equation implies that the time \( t \) must be a scalar quantity, which means the right-hand side must also be a scalar.

### Step 5: Analyze the relationship of the vectors
For the two particles to collide, the vectors must satisfy the condition that the direction of \( \mathbf{r_1} - \mathbf{r_2} \) is the same as the direction of \( \mathbf{v_2} - \mathbf{v_1} \). This can be expressed as:
\[ \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|} = \frac{\mathbf{v_2} - \mathbf{v_1}}{|\mathbf{v_2} - \mathbf{v_1}|} \]

### Conclusion
The final relationship that must hold for the particles to collide is:
\[ \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|} = \frac{\mathbf{v_2} - \mathbf{v_1}}{|\mathbf{v_2} - \mathbf{v_1}|} \]