If the position vector `vec(a)` of the point (5, n) is such that `|vec(a)|=13`, then the value/values of n can be

Options

A. `+-8`
B. `+- 12`
C. only 8
D. only 12

Answer

To solve the problem, we need to determine the value(s) of \( n \) given that the position vector \( \vec{a} \) of the point (5, n) has a magnitude of 13.

### Step-by-step Solution:

1. **Write the position vector**:
The position vector \( \vec{a} \) for the point (5, n) can be expressed as:
\[ \vec{a} = 5 \hat{i} + n \hat{j} \]

2. **Calculate the magnitude of the position vector**:
The magnitude of the vector \( \vec{a} \) is given by:
\[ |\vec{a}| = \sqrt{(5)^2 + (n)^2} \]
This simplifies to:
\[ |\vec{a}| = \sqrt{25 + n^2} \]

3. **Set the magnitude equal to 13**:
According to the problem, the magnitude of the vector is 13. Therefore, we can set up the equation:
\[ \sqrt{25 + n^2} = 13 \]

4. **Square both sides to eliminate the square root**:
Squaring both sides gives:
\[ 25 + n^2 = 169 \]

5. **Rearrange the equation to solve for \( n^2 \)**:
Subtract 25 from both sides:
\[ n^2 = 169 - 25 \]
This simplifies to:
\[ n^2 = 144 \]

6. **Take the square root of both sides**:
To find \( n \), take the square root:
\[ n = \pm \sqrt{144} \]
Therefore:
\[ n = \pm 12 \]

### Conclusion:
The possible values of \( n \) are:
\[ n = 12 \quad \text{or} \quad n = -12 \]