If the angle between the vectors `hat(i)- m hat(j) and hat(j) + hat(k)` is `(pi)/(3)`, then what is the value of m?

Options

A. 0
B. 2
C. `-2`
D. None of these

Answer

To solve the problem, we need to find the value of \( m \) given that the angle between the vectors \( \hat{i} - m \hat{j} \) and \( \hat{j} + \hat{k} \) is \( \frac{\pi}{3} \).

### Step-by-Step Solution:

1. **Identify the Vectors**:
Let vector \( \mathbf{A} = \hat{i} - m \hat{j} \) and vector \( \mathbf{B} = \hat{j} + \hat{k} \).

2. **Use the Cosine Formula**:
The cosine of the angle \( \theta \) between two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]
We know \( \theta = \frac{\pi}{3} \), so \( \cos \frac{\pi}{3} = \frac{1}{2} \).

3. **Calculate the Dot Product**:
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (\hat{i} - m \hat{j}) \cdot (\hat{j} + \hat{k}) = \hat{i} \cdot \hat{j} + \hat{i} \cdot \hat{k} - m \hat{j} \cdot \hat{j} - m \hat{j} \cdot \hat{k} \]
Since \( \hat{i} \cdot \hat{j} = 0 \), \( \hat{i} \cdot \hat{k} = 0 \), \( \hat{j} \cdot \hat{j} = 1 \), and \( \hat{j} \cdot \hat{k} = 0 \), we have:
\[ \mathbf{A} \cdot \mathbf{B} = 0 + 0 - m(1) - 0 = -m \]

4. **Calculate the Magnitudes**:
The magnitude of \( \mathbf{A} \):
\[ |\mathbf{A}| = \sqrt{1^2 + (-m)^2} = \sqrt{1 + m^2} \]
The magnitude of \( \mathbf{B} \):
\[ |\mathbf{B}| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{2} \]

5. **Set Up the Equation**:
Substitute the values into the cosine formula:
\[ \frac{1}{2} = \frac{-m}{\sqrt{1 + m^2} \cdot \sqrt{2}} \]
Rearranging gives:
\[ -m = \frac{1}{2} \cdot \sqrt{1 + m^2} \cdot \sqrt{2} \]
Simplifying further:
\[ -m = \frac{\sqrt{2}}{2} \sqrt{1 + m^2} \]

6. **Square Both Sides**:
Squaring both sides to eliminate the square root:
\[ m^2 = \left(\frac{\sqrt{2}}{2} \sqrt{1 + m^2}\right)^2 \]
This expands to:
\[ m^2 = \frac{2}{4}(1 + m^2) = \frac{1}{2}(1 + m^2) \]

7. **Rearranging the Equation**:
Multiply both sides by 2 to eliminate the fraction:
\[ 2m^2 = 1 + m^2 \]
Rearranging gives:
\[ 2m^2 - m^2 = 1 \implies m^2 = 1 \]

8. **Find the Values of \( m \)**:
Taking the square root gives:
\[ m = \pm 1 \]

9. **Check the Options**:
The options provided are \( 0, 2, -2, \) and \( \text{none of these} \). Since \( m = 1 \) or \( m = -1 \) is not listed, the answer is \( \text{none of these} \).

### Final Answer:
The value of \( m \) is \( \pm 1 \), and the answer is **none of these**.