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?

Options

A. 4m
B. 4.5 m
C. 5m
D. 3m

Answer

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

1. **Identify the Position Vectors**:
The position vectors of the two balls are given as:
\[ \vec{r}_1 = 2 \, \text{m} \, \hat{i} + 7 \, \text{m} \, \hat{j} \]
\[ \vec{r}_2 = -2 \, \text{m} \, \hat{i} + 4 \, \text{m} \, \hat{j} \]

2. **Calculate the Displacement Vector**:
The displacement vector \(\vec{d}\) from ball 1 to ball 2 can be calculated using:
\[ \vec{d} = \vec{r}_2 - \vec{r}_1 \]
Substituting the position vectors:
\[ \vec{d} = (-2 \, \hat{i} + 4 \, \hat{j}) - (2 \, \hat{i} + 7 \, \hat{j}) \]
Simplifying this gives:
\[ \vec{d} = -2 \, \hat{i} + 4 \, \hat{j} - 2 \, \hat{i} - 7 \, \hat{j} \]
\[ \vec{d} = (-2 - 2) \, \hat{i} + (4 - 7) \, \hat{j} \]
\[ \vec{d} = -4 \, \hat{i} - 3 \, \hat{j} \]

3. **Calculate the Magnitude of the Displacement Vector**:
The magnitude of the displacement vector \(\vec{d}\) is given by:
\[ |\vec{d}| = \sqrt{(-4)^2 + (-3)^2} \]
Calculating the squares:
\[ |\vec{d}| = \sqrt{16 + 9} \]
\[ |\vec{d}| = \sqrt{25} \]
\[ |\vec{d}| = 5 \, \text{m} \]

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