The components of a vector along the x- and y- directions are `(n+1)` and 1, respectively. If the coordinate system is rotated by an angle `theta=60^(@)`, then the components change to `n` and 3. The value of `n` is

Options

A. 2
B. `cos 60^(@)`
C. `sin 60^(@)`
D. 3.5

Explanation

To solve the problem step by step, we will follow the information provided in the question and apply the concepts of vector components and rotation.

### Step 1: Identify the initial components of the vector
The components of the vector along the x- and y-directions are given as:
- \( x_1 = n + 1 \)
- \( y_1 = 1 \)

Thus, we can represent the vector as:
\[ \vec{V} = (n + 1) \hat{i} + 1 \hat{j} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of the vector \(\vec{V}\) can be calculated using the formula:
\[ |\vec{V}| = \sqrt{(x_1)^2 + (y_1)^2} \]
Substituting the values:
\[ |\vec{V}| = \sqrt{(n + 1)^2 + 1^2} = \sqrt{(n + 1)^2 + 1} \]

### Step 3: Identify the new components after rotation
After rotating the coordinate system by an angle \(\theta = 60^\circ\), the new components of the vector are given as:
- \( x_2 = n \)
- \( y_2 = 3 \)

### Step 4: Calculate the magnitude of the vector after rotation
The magnitude of the vector after rotation can also be calculated as:
\[ |\vec{V}| = \sqrt{(x_2)^2 + (y_2)^2} \]
Substituting the new values:
\[ |\vec{V}| = \sqrt{n^2 + 3^2} = \sqrt{n^2 + 9} \]

### Step 5: Set the magnitudes equal to each other
Since the magnitude of the vector remains the same before and after rotation, we can set the two expressions for magnitude equal to each other:
\[ \sqrt{(n + 1)^2 + 1} = \sqrt{n^2 + 9} \]

### Step 6: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (n + 1)^2 + 1 = n^2 + 9 \]

### Step 7: Expand and simplify the equation
Expanding the left side:
\[ n^2 + 2n + 1 + 1 = n^2 + 9 \]
This simplifies to:
\[ n^2 + 2n + 2 = n^2 + 9 \]

### Step 8: Cancel \(n^2\) from both sides
Subtract \(n^2\) from both sides:
\[ 2n + 2 = 9 \]

### Step 9: Solve for \(n\)
Subtract 2 from both sides:
\[ 2n = 7 \]
Now divide by 2:
\[ n = \frac{7}{2} \]

### Final Answer
The value of \(n\) is:
\[ \boxed{\frac{7}{2}} \]