The one of the value of x for which the angle between `c=x""i+j+k" and "d=i+xj+k" is "pi//3` is

Options

A. `1+sqrt(2)`
B. `2+sqrt(2)`
C. `3+sqrt(2)`
D. none of these

Answer

To solve the problem, we need to find the value of \( x \) for which the angle between the vectors \( \mathbf{c} = x\mathbf{i} + \mathbf{j} + \mathbf{k} \) and \( \mathbf{d} = \mathbf{i} + x\mathbf{j} + \mathbf{k} \) is \( \frac{\pi}{3} \).

### Step-by-Step Solution:

1. **Identify the vectors**:
\[ \mathbf{c} = x\mathbf{i} + \mathbf{j} + \mathbf{k} \]
\[ \mathbf{d} = \mathbf{i} + x\mathbf{j} + \mathbf{k} \]

2. **Use the formula for the dot product**:
The angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \]
Here, \( \theta = \frac{\pi}{3} \), so \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

3. **Calculate the dot product \( \mathbf{c} \cdot \mathbf{d} \)**:
\[ \mathbf{c} \cdot \mathbf{d} = (x\mathbf{i} + \mathbf{j} + \mathbf{k}) \cdot (\mathbf{i} + x\mathbf{j} + \mathbf{k}) \]
\[ = x \cdot 1 + 1 \cdot x + 1 \cdot 1 = x + x + 1 = 2x + 1 \]

4. **Calculate the magnitudes of \( \mathbf{c} \) and \( \mathbf{d} \)**:
\[ |\mathbf{c}| = \sqrt{x^2 + 1^2 + 1^2} = \sqrt{x^2 + 2} \]
\[ |\mathbf{d}| = \sqrt{1^2 + x^2 + 1^2} = \sqrt{x^2 + 2} \]

5. **Substitute into the dot product formula**:
\[ 2x + 1 = |\mathbf{c}| |\mathbf{d}| \cos\left(\frac{\pi}{3}\right) \]
\[ 2x + 1 = \sqrt{x^2 + 2} \cdot \sqrt{x^2 + 2} \cdot \frac{1}{2} \]
\[ 2x + 1 = \frac{(x^2 + 2)}{2} \]

6. **Multiply both sides by 2 to eliminate the fraction**:
\[ 4x + 2 = x^2 + 2 \]

7. **Rearrange the equation**:
\[ x^2 - 4x + 2 = 0 \]

8. **Use the quadratic formula to solve for \( x \)**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -4, c = 2 \):
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
\[ = \frac{4 \pm \sqrt{16 - 8}}{2} \]
\[ = \frac{4 \pm \sqrt{8}}{2} \]
\[ = \frac{4 \pm 2\sqrt{2}}{2} \]
\[ = 2 \pm \sqrt{2} \]

### Final Values:
Thus, the possible values of \( x \) are:
\[ x = 2 + \sqrt{2} \quad \text{and} \quad x = 2 - \sqrt{2} \]