The vector `hati+xhatj+3hatk` is rotated through an angle `theta` and doubled in magnitude then it becomes `4hati+(4x-2)hatj+2hatk`. The values of `x` are

Options

A. `-2/3`
B. `1/3`
C. `2/3`
D. `2`

Answer

To solve the problem, we need to find the value of \( x \) given that the vector \( \hat{i} + x \hat{j} + 3 \hat{k} \) is rotated through an angle \( \theta \) and then doubled in magnitude, resulting in the vector \( 4 \hat{i} + (4x - 2) \hat{j} + 2 \hat{k} \).

### Step-by-Step Solution:

1. **Write the initial vector and the final vector:**
- Initial vector: \( \mathbf{A} = \hat{i} + x \hat{j} + 3 \hat{k} \)
- Final vector after rotation and doubling: \( \mathbf{B} = 4 \hat{i} + (4x - 2) \hat{j} + 2 \hat{k} \)

2. **Calculate the magnitude of the initial vector:**
\[ |\mathbf{A}| = \sqrt{1^2 + x^2 + 3^2} = \sqrt{1 + x^2 + 9} = \sqrt{x^2 + 10} \]

3. **Double the magnitude of the initial vector:**
\[ 2 |\mathbf{A}| = 2 \sqrt{x^2 + 10} \]

4. **Calculate the magnitude of the final vector:**
\[ |\mathbf{B}| = \sqrt{4^2 + (4x - 2)^2 + 2^2} = \sqrt{16 + (4x - 2)^2 + 4} \]
\[ = \sqrt{20 + (4x - 2)^2} \]

5. **Set the magnitudes equal to each other:**
Since the final vector is obtained by doubling the magnitude of the initial vector, we have:
\[ 2 \sqrt{x^2 + 10} = \sqrt{20 + (4x - 2)^2} \]

6. **Square both sides to eliminate the square roots:**
\[ (2 \sqrt{x^2 + 10})^2 = ( \sqrt{20 + (4x - 2)^2})^2 \]
\[ 4(x^2 + 10) = 20 + (4x - 2)^2 \]

7. **Expand the equation:**
\[ 4x^2 + 40 = 20 + (16x^2 - 16x + 4) \]
\[ 4x^2 + 40 = 20 + 16x^2 - 16x + 4 \]
\[ 4x^2 + 40 = 16x^2 - 16x + 24 \]

8. **Rearrange the equation:**
\[ 0 = 16x^2 - 4x^2 - 16x + 24 - 40 \]
\[ 0 = 12x^2 - 16x - 16 \]

9. **Divide the entire equation by 4:**
\[ 0 = 3x^2 - 4x - 4 \]

10. **Use the quadratic formula to solve for \( x \):**
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -4 \), and \( c = -4 \):
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3} \]
\[ = \frac{4 \pm \sqrt{16 + 48}}{6} \]
\[ = \frac{4 \pm \sqrt{64}}{6} \]
\[ = \frac{4 \pm 8}{6} \]

11. **Calculate the two possible values for \( x \):**
- First solution:
\[ x = \frac{12}{6} = 2 \]
- Second solution:
\[ x = \frac{-4}{6} = -\frac{2}{3} \]

### Final Values of \( x \):
The values of \( x \) are \( 2 \) and \( -\frac{2}{3} \).