The position vectors of two balls are given by
`vec(r )_(1)=2 (m)i+7(m)j`
`vec(r )_(2)= -2(m)i+4(m)j`
What will be the distance between the two balls?

Explanation

To find the distance between the two balls given their position vectors, we can follow these steps:

### Step 1: Write down the position vectors
The position vectors of the two balls are given as:
- \(\vec{r}_1 = 2 \, \text{m} \, \hat{i} + 7 \, \text{m} \, \hat{j}\)
- \(\vec{r}_2 = -2 \, \text{m} \, \hat{i} + 4 \, \text{m} \, \hat{j}\)

### Step 2: Find the displacement vector
The displacement vector \(\vec{R}\) between the two balls can be calculated using the formula:
\[ \vec{R} = \vec{r}_2 - \vec{r}_1 \]
Substituting the values:
\[ \vec{R} = (-2 \, \hat{i} + 4 \, \hat{j}) - (2 \, \hat{i} + 7 \, \hat{j}) \]
This simplifies to:
\[ \vec{R} = -2 \, \hat{i} + 4 \, \hat{j} - 2 \, \hat{i} - 7 \, \hat{j} \]
Combining like terms:
\[ \vec{R} = (-2 - 2) \, \hat{i} + (4 - 7) \, \hat{j} = -4 \, \hat{i} - 3 \, \hat{j} \]

### Step 3: Calculate the magnitude of the displacement vector
The distance \(d\) between the two balls is the magnitude of the displacement vector \(\vec{R}\):
\[ d = |\vec{R}| = \sqrt{(-4)^2 + (-3)^2} \]
Calculating the squares:
\[ d = \sqrt{16 + 9} = \sqrt{25} \]
Thus, the distance is:
\[ d = 5 \, \text{m} \]

### Conclusion
The distance between the two balls is \(5 \, \text{m}\).

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