The vector `hati+xhatj+3hatk` is rotated through an angle `theta` and doubled in magnitude, then it becomes `4hati+(4x-2)hatj+2hatk`. Then values of x are (A) `-2/3` (B) `1/3` (C) `2/3` (D) 2

Options

A. `1`
B. `-2//3`
C. `2`
D. `4//3`

Explanation

To solve the problem, we need to find the value of \( x \) given the vectors and their conditions. Let's break it down step by step.

### Step 1: Define the Vectors
We are given two vectors:
1. \( \vec{A} = \hat{i} + x\hat{j} + 3\hat{k} \)
2. After rotation and doubling the magnitude, we have \( \vec{B} = 4\hat{i} + (4x - 2)\hat{j} + 2\hat{k} \)

### Step 2: Calculate the Magnitude of Vector A
The magnitude of vector \( \vec{A} \) is calculated using the formula:
\[ |\vec{A}| = \sqrt{(1)^2 + (x)^2 + (3)^2} = \sqrt{1 + x^2 + 9} = \sqrt{x^2 + 10} \]

### Step 3: Calculate the Magnitude of Vector B
The magnitude of vector \( \vec{B} \) is calculated as follows:
\[ |\vec{B}| = \sqrt{(4)^2 + (4x - 2)^2 + (2)^2} = \sqrt{16 + (4x - 2)^2 + 4} \]
Expanding \( (4x - 2)^2 \):
\[ (4x - 2)^2 = 16x^2 - 16x + 4 \]
So, the magnitude becomes:
\[ |\vec{B}| = \sqrt{16 + 16x^2 - 16x + 4} = \sqrt{16x^2 - 16x + 20} \]

### Step 4: Set Up the Equation
According to the problem, the magnitude of vector \( \vec{B} \) is double that of vector \( \vec{A} \):
\[ |\vec{B}| = 2|\vec{A}| \]
Substituting the magnitudes we calculated:
\[ \sqrt{16x^2 - 16x + 20} = 2\sqrt{x^2 + 10} \]

### Step 5: Square Both Sides
To eliminate the square roots, we square both sides:
\[ 16x^2 - 16x + 20 = 4(x^2 + 10) \]
Expanding the right side:
\[ 16x^2 - 16x + 20 = 4x^2 + 40 \]

### Step 6: Rearranging the Equation
Now, we rearrange the equation:
\[ 16x^2 - 4x^2 - 16x + 20 - 40 = 0 \]
This simplifies to:
\[ 12x^2 - 16x - 20 = 0 \]

### Step 7: Simplifying the Quadratic Equation
We can simplify this equation by dividing everything by 4:
\[ 3x^2 - 4x - 5 = 0 \]

### Step 8: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3, b = -4, c = -5 \):
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot (-5)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ x = \frac{4 \pm \sqrt{16 + 60}}{6} = \frac{4 \pm \sqrt{76}}{6} = \frac{4 \pm 2\sqrt{19}}{6} = \frac{2 \pm \sqrt{19}}{3} \]

### Step 9: Finding the Values of x
Calculating the two possible values:
1. \( x_1 = \frac{2 + \sqrt{19}}{3} \)
2. \( x_2 = \frac{2 - \sqrt{19}}{3} \)

### Step 10: Check Against Options
Now we check which of the options (A) \(-\frac{2}{3}\), (B) \(\frac{1}{3}\), (C) \(\frac{2}{3}\), (D) \(2\) match our results.

### Final Answer
The values of \( x \) that satisfy the equation are:
- \( x = 2 \) (Option D)
- \( x = -\frac{2}{3} \) (Option A)

Thus, the correct options are (A) \(-\frac{2}{3}\) and (D) \(2\).