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

Explanation

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 is given by:
\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]
Here, \( \theta = \frac{\pi}{3} \), so:
\[ \cos \frac{\pi}{3} = \frac{1}{2} \]

3. **Calculate the Dot Product \( \mathbf{a} \cdot \mathbf{b} \):**
\[ \mathbf{a} \cdot \mathbf{b} = (\hat{i} - m\hat{j}) \cdot (\hat{j} + \hat{k}) = \hat{i} \cdot \hat{j} + \hat{i} \cdot \hat{k} - m\hat{j} \cdot \hat{j} - m\hat{j} \cdot \hat{k} \]
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 of the Vectors:**
- For \( \mathbf{a} \):
\[ |\mathbf{a}| = \sqrt{1^2 + (-m)^2} = \sqrt{1 + m^2} \]
- For \( \mathbf{b} \):
\[ |\mathbf{b}| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{2} \]

5. **Substitute into the Cosine Formula:**
Now substituting into the cosine formula:
\[ \frac{1}{2} = \frac{-m}{\sqrt{1 + m^2} \cdot \sqrt{2}} \]

6. **Cross-Multiply:**
Cross-multiplying gives:
\[ 1 \cdot \sqrt{1 + m^2} \cdot \sqrt{2} = -2m \]
Simplifying this:
\[ \sqrt{2(1 + m^2)} = -2m \]

7. **Square Both Sides:**
Squaring both sides results in:
\[ 2(1 + m^2) = 4m^2 \]
Expanding gives:
\[ 2 + 2m^2 = 4m^2 \]

8. **Rearranging the Equation:**
Rearranging leads to:
\[ 2 = 4m^2 - 2m^2 \]
\[ 2 = 2m^2 \]
Dividing both sides by 2:
\[ 1 = m^2 \]

9. **Finding the Value of \( m \):**
Taking the square root gives:
\[ m = \pm 1 \]

### Final Answer:
The values of \( m \) are \( 1 \) and \( -1 \).