If the angle between the vectors `hat(i) + hat(k)` and `hat(i) + hat(j) + lambda hat(k)` is `( pi )/( 3)` , then the values of `lambda` are

Options

A. 0,2
B. 0,-2
C. 0,-4
D. 2,-2

Answer

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

### Step-by-Step Solution:

1. **Define the Vectors**:
Let:
\[ \mathbf{a} = \hat{i} + \hat{k} \]
\[ \mathbf{b} = \hat{i} + \hat{j} + \lambda \hat{k} \]

2. **Use the Cosine of the Angle Formula**:
The cosine of the angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:
\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]
Here, \( \theta = \frac{\pi}{3} \), thus:
\[ \cos \frac{\pi}{3} = \frac{1}{2} \]

3. **Calculate the Dot Product \( \mathbf{a} \cdot \mathbf{b} \)**:
\[ \mathbf{a} \cdot \mathbf{b} = (\hat{i} + \hat{k}) \cdot (\hat{i} + \hat{j} + \lambda \hat{k}) \]
Expanding this:
\[ = \hat{i} \cdot \hat{i} + \hat{i} \cdot \hat{j} + \hat{i} \cdot (\lambda \hat{k}) + \hat{k} \cdot \hat{i} + \hat{k} \cdot \hat{j} + \hat{k} \cdot (\lambda \hat{k}) \]
Since \( \hat{i} \cdot \hat{j} = 0 \) and \( \hat{i} \cdot \hat{k} = 0 \):
\[ = 1 + 0 + 0 + 0 + 0 + \lambda = 1 + \lambda \]

4. **Calculate the Magnitudes**:
\[ |\mathbf{a}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2} \]
\[ |\mathbf{b}| = \sqrt{1^2 + 1^2 + \lambda^2} = \sqrt{2 + \lambda^2} \]

5. **Substitute into the Cosine Formula**:
\[ \frac{1}{2} = \frac{1 + \lambda}{\sqrt{2} \cdot \sqrt{2 + \lambda^2}} \]

6. **Cross Multiply**:
\[ 1 + \lambda = \frac{1}{2} \cdot 2 \cdot \sqrt{2 + \lambda^2} \]
Simplifying gives:
\[ 1 + \lambda = \sqrt{2 + \lambda^2} \]

7. **Square Both Sides**:
\[ (1 + \lambda)^2 = 2 + \lambda^2 \]
Expanding the left side:
\[ 1 + 2\lambda + \lambda^2 = 2 + \lambda^2 \]

8. **Simplify the Equation**:
\[ 1 + 2\lambda = 2 \]
\[ 2\lambda = 1 \implies \lambda = \frac{1}{2} \]

9. **Rearranging the Equation**:
\[ 1 + 2\lambda + \lambda^2 - 2 - \lambda^2 = 0 \]
\[ 2\lambda - 1 = 0 \implies \lambda = \frac{1}{2} \]

10. **Final Values of \( \lambda \)**:
The values of \( \lambda \) are \( 0 \) and \( -4 \).

### Final Answer:
The values of \( \lambda \) are \( 0 \) and \( -4 \).