Two vector `2hat(i)+m hat(j)-3 n hat(k) and 5 hat(i) + 3 m hat(j)+ n hat(k)` are such that their magnitudes are respectively `sqrt(14) and sqrt(35)`, where m, n are integers. Which one of the following is correct?

Options

A. m takes 1 value, n takes 1 value
B. m takes 1 value, n takes 2 values
C. m takes 2 value, n takes 1 value
D. m takes 2 value, n takes 2 values

Explanation

To solve the problem, we need to find the values of integers \( m \) and \( n \) such that the magnitudes of the given vectors are satisfied.

### Step-by-Step Solution:

1. **Magnitude of the First Vector**:
The first vector is \( \mathbf{A} = 2\hat{i} + m\hat{j} - 3n\hat{k} \).
The magnitude of vector \( \mathbf{A} \) is given by:
\[ |\mathbf{A}| = \sqrt{(2)^2 + (m)^2 + (-3n)^2} = \sqrt{4 + m^2 + 9n^2} \]
We know that this magnitude equals \( \sqrt{14} \):
\[ \sqrt{4 + m^2 + 9n^2} = \sqrt{14} \]
Squaring both sides, we get:
\[ 4 + m^2 + 9n^2 = 14 \]
Simplifying this gives us:
\[ m^2 + 9n^2 = 10 \quad \text{(Equation 1)} \]

2. **Magnitude of the Second Vector**:
The second vector is \( \mathbf{B} = 5\hat{i} + 3m\hat{j} + n\hat{k} \).
The magnitude of vector \( \mathbf{B} \) is given by:
\[ |\mathbf{B}| = \sqrt{(5)^2 + (3m)^2 + (n)^2} = \sqrt{25 + 9m^2 + n^2} \]
We know that this magnitude equals \( \sqrt{35} \):
\[ \sqrt{25 + 9m^2 + n^2} = \sqrt{35} \]
Squaring both sides, we get:
\[ 25 + 9m^2 + n^2 = 35 \]
Simplifying this gives us:
\[ 9m^2 + n^2 = 10 \quad \text{(Equation 2)} \]

3. **Solving the System of Equations**:
We now have two equations:
- \( m^2 + 9n^2 = 10 \) (Equation 1)
- \( 9m^2 + n^2 = 10 \) (Equation 2)

To solve these equations, we can manipulate them. Let's subtract Equation 1 from Equation 2:
\[ (9m^2 + n^2) - (m^2 + 9n^2) = 10 - 10 \]
This simplifies to:
\[ 8m^2 - 8n^2 = 0 \]
Dividing by 8 gives:
\[ m^2 = n^2 \]
This implies:
\[ m = n \quad \text{or} \quad m = -n \]

4. **Finding Integer Solutions**:
Since \( m \) and \( n \) are integers, we can substitute \( n = m \) and \( n = -m \) into either Equation 1 or Equation 2 to find possible integer values.

Substituting \( n = m \) into Equation 1:
\[ m^2 + 9m^2 = 10 \implies 10m^2 = 10 \implies m^2 = 1 \implies m = 1 \text{ or } m = -1 \]
Thus, \( n = 1 \) or \( n = -1 \).

Substituting \( n = -m \) into Equation 1:
\[ m^2 + 9(-m)^2 = 10 \implies m^2 + 9m^2 = 10 \implies 10m^2 = 10 \implies m^2 = 1 \implies m = 1 \text{ or } m = -1 \]
Thus, \( n = -1 \) or \( n = 1 \).

5. **Conclusion**:
Both \( m \) and \( n \) can take the values \( 1 \) or \( -1 \). Therefore, \( m \) takes 2 values and \( n \) takes 2 values.

### Final Answer:
Both \( m \) and \( n \) can take 2 values each.