P is the point `vec(i)+x vec(j)+3vec(k)`. The vector `bar(OP)` ('O' is the origin) is rotated about the point 'O' through an angle `theta`. Q is point `4vec(i)+(4x-2)vec(j)+2vec(k)` on the new support of `bar(OP)` such that `OQ=2OP`. Then x value is

Options

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

Answer

To solve the problem step by step, we need to analyze the given information carefully.

### Step 1: Define the position vector of point P
The position vector of point P is given as:
\[ \vec{P} = \vec{i} + x \vec{j} + 3 \vec{k} \]
Thus, the vector \(\vec{OP}\) from the origin O to point P can be expressed as:
\[ \vec{OP} = \vec{i} + x \vec{j} + 3 \vec{k} \]

### Step 2: Calculate the length of vector OP
The length of vector \(\vec{OP}\) can be calculated using the formula for the magnitude of a vector:
\[ |\vec{OP}| = \sqrt{(1)^2 + (x)^2 + (3)^2} = \sqrt{1 + x^2 + 9} = \sqrt{x^2 + 10} \]

### Step 3: Define the position vector of point Q
The position vector of point Q is given as:
\[ \vec{Q} = 4\vec{i} + (4x - 2)\vec{j} + 2\vec{k} \]
Thus, the vector \(\vec{OQ}\) from the origin O to point Q can be expressed as:
\[ \vec{OQ} = 4\vec{i} + (4x - 2)\vec{j} + 2\vec{k} \]

### Step 4: Calculate the length of vector OQ
The length of vector \(\vec{OQ}\) is given by:
\[ |\vec{OQ}| = \sqrt{(4)^2 + (4x - 2)^2 + (2)^2} = \sqrt{16 + (4x - 2)^2 + 4} \]
Simplifying this gives:
\[ |\vec{OQ}| = \sqrt{20 + (4x - 2)^2} \]

### Step 5: Set up the equation based on the condition OQ = 2OP
According to the problem, we have:
\[ |\vec{OQ}| = 2 |\vec{OP}| \]
Substituting the lengths we calculated:
\[ \sqrt{20 + (4x - 2)^2} = 2\sqrt{x^2 + 10} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 20 + (4x - 2)^2 = 4(x^2 + 10) \]

### Step 7: Expand and simplify the equation
Expanding both sides:
\[ 20 + (16x^2 - 16x + 4) = 4x^2 + 40 \]
This simplifies to:
\[ 16x^2 - 16x + 24 = 4x^2 + 40 \]
Rearranging gives:
\[ 12x^2 - 16x - 16 = 0 \]

### Step 8: Divide the equation by 4 for simplicity
Dividing the entire equation by 4:
\[ 3x^2 - 4x - 4 = 0 \]

### Step 9: Use the quadratic formula to find x
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 3\), \(b = -4\), and \(c = -4\):
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ x = \frac{4 \pm \sqrt{16 + 48}}{6} = \frac{4 \pm \sqrt{64}}{6} = \frac{4 \pm 8}{6} \]
Thus, we have two potential solutions:
\[ x = \frac{12}{6} = 2 \quad \text{and} \quad x = \frac{-4}{6} = -\frac{2}{3} \]

### Final Result
The values of \(x\) are:
\[ x = 2 \quad \text{or} \quad x = -\frac{2}{3} \]